Extrinsic Geometry and Linear Differential Equations

We give a unified method for the general equivalence problem of extrinsic geometry, based on our formulation of a general extrinsic geometry as that of an osculating map : ( , f) → / ⁰ ⊂ Flag( , ) from a filtered manifold ( , f) to a homogeneous space / ⁰ in a flag variety Flag( , ), where L is...

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Veröffentlicht in:	Symmetry, Integrability and Geometry: Methods and Applications
Datum:	2021
Hauptverfasser:	Doubrov, Boris, Machida, Yoshinori, Morimoto, Tohru
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Sprache:	Englisch
Veröffentlicht:	Інститут математики НАН України 2021
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Zitieren:	Extrinsic Geometry and Linear Differential Equations. Boris Doubrov, Yoshinori Machida and Tohru Morimoto. SIGMA 17 (2021), 061, 60 pages

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author	Doubrov, Boris Machida, Yoshinori Morimoto, Tohru
author_facet	Doubrov, Boris Machida, Yoshinori Morimoto, Tohru
citation_txt	Extrinsic Geometry and Linear Differential Equations. Boris Doubrov, Yoshinori Machida and Tohru Morimoto. SIGMA 17 (2021), 061, 60 pages
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container_title	Symmetry, Integrability and Geometry: Methods and Applications
description	We give a unified method for the general equivalence problem of extrinsic geometry, based on our formulation of a general extrinsic geometry as that of an osculating map : ( , f) → / ⁰ ⊂ Flag( , ) from a filtered manifold ( , f) to a homogeneous space / ⁰ in a flag variety Flag( , ), where L is a finite-dimensional Lie group and ⁰ its closed subgroup. We establish an algorithm to obtain complete systems of invariants for the osculating maps that satisfy a reasonable regularity condition, namely, a constant symbol of type ( ₋, gr , ). We show the categorical isomorphism between the extrinsic geometries in flag varieties and the (weighted) involutive systems of linear differential equations of finite type. Therefore, we also obtain a complete system of invariants for a general involutive system of linear differential equations of finite type and of constant symbol. The invariants of an osculating map (or an involutive system of linear differential equations) are proved to be controlled by the cohomology group ¹₊( ₋, / ¯) which is defined algebraically from the symbol of the osculating map (resp. involutive system), and which, in many cases (in particular, if the symbol is associated with a simple Lie algebra and its irreducible representation), can be computed by the algebraic harmonic theory, and the vanishing of which gives rigidity theorems in various concrete geometries. We also extend the theory to the case when is infinite-dimensional.
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 061, 60 pages Extrinsic Geometry and Linear Differential Equations Boris DOUBROV a, Yoshinori MACHIDA b and Tohru MORIMOTO cd a) Faculty of Mathematics and Mechanics, Belarusian State University, Nezavisimosti ave. 4, Minsk 220030, Belarus E-mail: doubrov@bsu.by b) Shizuoka University, Shizuoka 422-8529, Japan E-mail: yomachi212@gmail.com c) Seki Kowa Institute of Mathematics, Yokkaichi University, Yokkaichi 512-8045, Japan d) Institut Kiyoshi Oka de Mathématiques, Nara Women’s University, Nara 630-8506, Japan E-mail: morimoto@cc.nara-wu.ac.jp Received June 04, 2020, in final form June 04, 2021; Published online June 17, 2021 https://doi.org/10.3842/SIGMA.2021.061 Abstract. We give a unified method for the general equivalence problem of extrinsic geo- metry, on the basis of our formulation of a general extrinsic geometry as that of an osculat- ing map ϕ : (M, f) → L/L0 ⊂ Flag(V, φ) from a filtered manifold (M, f) to a homogeneous space L/L0 in a flag variety Flag(V, φ), where L is a finite-dimensional Lie group and L0 its closed subgroup. We establish an algorithm to obtain the complete systems of invariants for the osculating maps which satisfy the reasonable regularity condition of constant symbol of type (g−, grV,L). We show the categorical isomorphism between the extrinsic geometries in flag varieties and the (weighted) involutive systems of linear differential equations of fi- nite type. Therefore we also obtain a complete system of invariants for a general involutive systems of linear differential equations of finite type and of constant symbol. The invariants of an osculating map (or an involutive system of linear differential equations) are proved to be controlled by the cohomology group H1 +(g−, l/ḡ) which is defined algebraically from the symbol of the osculating map (resp. involutive system), and which, in many cases (in partic- ular, if the symbol is associated with a simple Lie algebra and its irreducible representation), can be computed by the algebraic harmonic theory, and the vanishing of which gives rigidity theorems in various concrete geometries. We also extend the theory to the case when L is infinite dimensional. Key words: extrinsic geometry; filtered manifold; flag variety; osculating map; involutive systems of linear differential equations; extrinsic Cartan connection; rigidity of rational homogeneous varieties 2020 Mathematics Subject Classification: 53A55; 53C24; 53C30; 53D10 Dedicated to Élie Cartan on the 150th anniversary of his birth 1 Introduction In geometry the distinction between intrinsic and extrinsic geometry is fundamental. The former treats spaces, while the latter figures. A space may be defined to be a set S equipped with a geometric structure γ on S, which can be defined by assigning a subset of certain associated set to S. A figure may be understood to be a subset of a space, or rather a map ϕ : X → A from a space X to a space A. Two spaces X = (SX , γX) and Y = (SY , γY ) are said to be (intrinsically) equivalent or iso- morphic if there exists a bijection f : SX → SY satisfying f∗γX = γY , where f∗ denotes the associated map to f . mailto:doubrov@bsu.by mailto:yomachi212@gmail.com mailto:morimoto@cc.nara-wu.ac.jp https://doi.org/10.3842/SIGMA.2021.061 2 B. Doubrov, Y. Machida and T. Morimoto Two figures ϕ : X → A and ψ : Y → B are said to be (extrinsically) equivalent or isomorphic if there exist isomorphisms of spaces f : X → Y and F : A → B such that F ◦ ϕ = ψ ◦ f . Without loss of generality, we may assume that the ambient spaces A and B coincide and equal to a homogeneous space L/L0 with a group L and its subgroup L0. Then the isomorphisms F : L/L0 → L/L0 should be understood to be the left translations Λa by a ∈ L. We may then say that intrinsic geometry studies those properties of spaces X that are invari- ant under intrinsic equivalences, and extrinsic geometry those of figures ϕ : X → L/L0 invariant under extrinsic equivalences. In both geometries one of the fundamental problems is the so-called equivalence problem, that is to find the criteria to determine whether two spaces or two figures are equivalent or not, which, in smooth or analytic category, leads to the problem of finding (the complete) differential invariants of a space or a figure arbitrary given. For intrinsic geometry we know now well established general theories which have been develo- ped by Lie, Klein, Cartan and then by many authors, in particular, in [3, 4, 20, 22, 32, 33, 35, 36]. For extrinsic geometry which has much longer history we know a great amount of works done in various concrete problems. However, the general theory does not seem to have been fully developed. The main purpose of the present paper is to develop a systematic study on extrinsic geometry in general setting of nilpotent geometry so as to be comparable to the theories in intrinsic geometry. The objects of extrinsic geometry with which we are mainly concerned and which are the most general by many reasons are the morphisms, or in geometrical terms, osculating maps, ϕ : (M, f)→ L/L0 ⊂ Flag(V, φ) from a filtered manifold (M, f) to a homogeneous space L/L0 in a flag variety Flag(V, φ). We shall then give an algorithm to find the complete invariants of the osculating maps ϕ under certain reasonable conditions. The framework developed here is a nilpotent generalization in extrinsic geometry of the Cartan method (of bundles) of moving frames, and is in good harmony with the one in intrinsic geometry developed in [22, 35, 36]. The two methods in intrinsic and extrinsic geometry thus integrated will give a unified view for clearer understanding of geometry as well as for applications to differential equations. Now let us explain the content of paper more precisely by following the sections. In Section 2 first we recall the notion of filtered manifold. It is a generalization and a refi- nement of that of usual manifold and is defined to be a smooth manifold M endowed with a tangential filtration f = {fp}p∈Z. At each point x ∈ M there is associated a nilpotent graded Lie algebra gr fx which, taking the place of usual tangent space, plays a fundamental role in the method of nilpotent approximation initiated by Tanaka [35, 36]. We then fix the notation for the flag variety Flag(V, φ) by defining it as the manifold con- sisting of all filtration of V isomorphic to a given filtration φ, where V is a vector space over R or C assumed to be finite dimensional unless otherwise stated, therefore it is represented as a homogeneous space Flag(V, φ) = GL(V )/GL(V )0, where GL(V )0 is the isotropy subgroup at φ. Moreover it is a filtered manifold with natural left invariant tangential filtration. Then we call a map ϕ : (M, f)→ L/L0 ⊂ Flag(V, φ) osculating if fp ϕq ⊂ ϕp+q for all p, q ∈ Z. This is equivalent to saying that the map ϕ : (M, f) → Flag(V, φ) is a morphism of filtered manifolds. Extrinsic Geometry and Linear Differential Equations 3 In Section 3 we introduce, for a given homogeneous space L/L0 ⊂ Flag(V, φ), three categories, that is, those of L/L0 extrinsic geometries, L/L0 differential equations and L/L0 extrinsic bundles. The first one is just the category whose objects are the osculating maps ϕ : (M, f)→ L/L0 ⊂ Flag(V, φ). The second one represents the category of (weighted) involutive systems of linear differential equations defined in weighted jet bundles (of finite type if V is finite dimensional). See [23] for the definition of (weighted) involutive systems. The third one is the category of the principal fibre bundles P over filtered manifolds (M, f) with structure group K0 ⊂ L0 equipped with an l valued 1-form ω satisfying above all dω + 1 2 [ω, ω] = 0 as well as some natural conditions with respect to the bundle structure. In each category we define the notion of a congruence class to clarify our implicit identifica- tion of those objects that are congruent. For instance, two extrinsic bundles ( P,K0, (M, f), ω ) and (P ′,K ′0, (M ′, f), ω′) are called congruent if the extensions of structure groups to L0 of P and P ′ are isomorphic with (M, f) = (M ′, f), the induced map on the base spaces being the identity map. In Section 4 we show that these three categories are categorically isomorphic up to congru- ences and coverings. Thus, roughly speaking, the extrinsic geometry in flag manifold and the geometry of involutive systems of linear differential equations are equivalent and their equiva- lence problem reduces to that of extrinsic bundles. In Section 5 we study the equivalence problem for a class of submanifolds in L/L0. The first basic invariant of an osculating map ϕ is grϕ called the symbol of ϕ. It assigns to x ∈ M the space grϕx that has a structure of graded gr fx-module. We say ϕ is of constant symbol of type (g−, grV,L) if (gr fx, grϕx) is L-isomorphic to (g−, grV ) for all x ∈ M , where g− is a graded nilpotent Lie subalgebra of gr− l = ⊕ p<0 grp l. We make the following assumption throughout: (C0) There exists a filtration preserving identification of V and grV that identifies l ⊂ gl(V ) and gr(l, φ) ⊂ gl(grV ). We then work in the subcategory of constant symbol of type (g−, grV,L). An algebraic aspect of this subcategory is represented by the graded Lie algebra ḡ = ⊕ ḡp ⊂ ⊕ lp which is called the relative (or extrinsic) prolongation of g− in l and is defined to be the maximal graded subalgebra of l whose negative part coincides with g−. The standard model in this category is given by ϕmodel : G/G 0 → L/L0, where G and G 0 are Lie subgroups corresponding to ḡ and ḡ0. The deviation of ϕ from ϕmodel is then measured by passing to the corresponding extrinsic bundle Q and by its structure function χ = ∑ i≥1 χi defined on Q. To go further we pose the following assumptions: (C1) There exists a Ḡ0-invariant graded subspace ḡ′ ⊂ l such that l = ḡ⊕ ḡ′. (C2) There exists a G 0 -invariant graded subspace W = ⊕ Wp ⊂ φ1 Hom(g−, l/ḡ) such that φ1 Hom(g−, l/ḡ) = ∂φ1(l/ḡ)⊕W . Here ∂ is the coboundary operator of the following differential complex equipped with the ∂-invariant induced filtration φ: 0→ l/ḡ ∂→ Hom(g−, l/ḡ) ∂→ Hom ( ∧2g−, l/ḡ ) ∂→ · · · , whose cohomology group H(g−, l/ḡ) then plays an important role. Now our main theorem may be stated as follows: 4 B. Doubrov, Y. Machida and T. Morimoto Theorem. Under the assumptions (C0), (C1) and (C2), for every L/L0 extrinsic bundle (Q,ωQ) (L/L0 extrinsic geometry or L/L0 differential equation) of constant symbol of type (g−, grV,L), we can construct canonically an extrinsic bundle (P, ωP ) such that (P, ωP ) is con- gruent to (Q,ωQ) and is a reduction to the subgroup Ḡ0 and the structure function χP of P takes its value in W . The extrinsic bundle (P, ω) thus constructed is called W -normal extrinsic Cartan connection and is governed by the structure equation dω + 1 2 [ω, ω] = 0, ω = ωI + ωII , ωII = χωI , where ωI , ωII denote the ḡ, ḡ′ components of ω respectively. It follows from the structure equation, in particular, that ∂χk = Ψk, where Ψk is a differential polynomial of {χi, i < k} with Ψ1 = 0. Thus the structure function χ = ∑ i≥1 χi, and hence the structure of (P, ω) is uniquely determined up to the cohomology group H1 +(g−, l/ḡ). Therefore as a corollary we have: Corollary (rigidity). Any ϕ : (M, f) → L/L0 of constant symbol of type (g−, grV,L) is locally equivalent to ϕmodel if (C0), (C1) and (C2) hold and if H1 +(g−, l/ḡ) = 0. In Section 6 we apply the above theorem to what may be called “extrinsic parabolic geo- metry”. Take a simple graded Lie algebra g and an irreducible representation g → gl(V ), and further take a graded subalgebra l = ⊕ p∈Z lp of gl(V ) containing g and let L and L0 be the Lie subgroup of GL(V ) corresponding to l and l0 = ⊕ p≥0 lp. The category of L/L0 extrinsic geometries of constant symbol of type (g−, grV,L) determined by the above data satisfies the assumptions (C0), (C1) and (C2): The relative prolongation ḡ of g− in l proves to be g itself or its central extension. Moreover there exist distinguished inner products on g and on V such that if we define the adjoint operator ∂∗ on the differential complex concerned by using the induced inner product then Ker ∂∗ will serve as W required in (C2). The standard model ϕmodel is an embedding of a rational homogeneous variety into a flag variety. Thus we have a large and rich class of L/L0 categories associated with each such (g, l, gl(V )) to which the theorem in Section 5 applies. In this case the cohomology group H1 +(g−, l/ḡ) can be computed by using root systems after Kostant’s method. We have carried computation for every simple graded Lie algebra g over C and its irreducible representation U to see when H1 +(g−, U) is trivial, which gives a lot of rigidity theorems and a rough picture about the extrinsic parabolic geometries. In Section 7 we further study the equivalence problem of extrinsic geometry in more general setting than in Section 5. Firstly, we treat the case where (C1) and (C2) are not necessarily satisfied. Then we can no more expect to have Cartan connections but we can still construct a series of bundles which give the complete invariants of a given osculating map ϕ and the invariants are again controlled by the cohomology group H1 +(g−, l/ḡ). Hence we see, in particular, that the corollary stated in Section 5 holds without the assumptions (C1) and (C2). Secondly, we generalize our theory to the case when L is infinite dimensional. The main motivation is, on one hand, to study extrinsic geometries with respect to infinite-dimensional transformation group such as the general diffeomorphism group or the contact transformation group, and, on the other hand, to study geometry of linear differential equations of infinite type. Extrinsic Geometry and Linear Differential Equations 5 Here we notice the remarkable similarity between intrinsic and extrinsic geometry which becomes visible and clear in this work: Recall that the equivalence problem of intrinsic geometry reduces to that of towers. A tower is a principal fibre bundle P with structure group G0 possibly infinite-dimensional equipped with a vector valued 1-form which defines an absolute parallelism of P satisfying certain natural conditions. While the equivalence problem of extrinsic geometry reduces to that of extrinsic bundles defined in this paper. In both intrinsic and extrinsic cases we are led to the equivalence problem of principal fibre bundles equipped with Pfaff systems. The constructing Cartan connections make nice harmony in intrinsic and extrinsic geometry, the condition (C) in intrinsic geometry [22] corresponding to (C2) in extrinsic geometry. Moreover, the prolongation and reduction method of Singer–Sternberg and Tanaka for intri- nsic geometry corresponds to the reduction method in Section 7 for extrinsic geometry. It is the first cohomology group H1 for extrinsic geometry and the second H2 for intrinsic geometry that pays the key role. Since extrinsic geometry has long history, there may be a lot of works which may re- late to our work. We cite here those works that have influenced us or that we find related to the present paper: Lie [18], Wilczynski [37, 38], Cartan [4, 5], Sulanke [34], Jensen [14], Kolář [15], Se-Ashi [30, 31], Sasaki [19, 27, 29], Sasaki–Yamaguchi–Yoshida [28], Fels–Olver [8], Hwang–Yamaguchi [12], Robles [26], Landsberg–Robles [17], Doubrov–Komrakov [6], Doubrov– Zelenko [7]. Among all we owe, in particular, to Wilczynski and Se-Ashi: At the beginning of twentieth century after the influence of Lie, Wilczynski developed a general theory of projective curves and ruled surfaces, recognizing that these geometrics are equivalent to those of ordinary linear differential equations and certain systems of linear partial differential equations of 2nd order. Se-Ashi reconstructed the differential invariants of ordinary linear differential equations by the method of bundles of moving frames, and found a prototype of extrinsic Cartan connections. Then he extended this construction to systems of linear differential equations associated with a simple graded Lie algebra of depth one and its irreducible representation. Our preset work may be viewed as a full generalization of the works of Wilczynski and Se-Ashi to the framework of nilpotent geometry and analysis in which we find all well unified. Even in extrinsic parabolic geometry, there are many interesting concrete examples. The sim- plest is the one associated with the simple Lie algebra sl(2) and its irreducible representation on Symn ( C2 ) , and this is exactly the geometry of curves in the projective space Pn or that one of linear ordinary differential equations of order n+1 treated by Wilczynski [37] and Se-Ashi [30, 31] (see also Ovsienko–Tabachnikov [25] for a modern exposition of Wilczynski results). The next interesting example which is non-trivial as nilpotent geometry is the one associated with (sl(3), Borel grading, adjoint representation) as shown by the works of Robles [26] and Landsberg–Robles [17]. We have carried a detailed study in this case after our general method, which will be an object of the second part of this paper. 2 Filtered manifolds and flag varieties 2.1 Filtered manifolds A filtered manifold is a smooth manifold M equipped with a tangential filtration f = {fp}p∈Z satisfying the following conditions: (i) Each fp is a subbundle of the tangent bundle TM and fp ⊃ fp+1. (ii) f0 = 0 and f−µ = TM for a non-negative integer µ. (iii) [fp, fq] ⊂ fp+q for p, q ∈ Z, where f• denotes the sheaf of germs of sections of f•. 6 B. Doubrov, Y. Machida and T. Morimoto The filtration {fp} is occasionally written as {fpTM} or as {T pM}. The fibre of fp at x ∈ M will be written as fpx, fpTxM , or T pxM . The filtered manifold will be denoted (M, f) or simply M . A (local) isomorphism ϕ : (M, f) → (M ′, f′) of two filtered manifolds is a (local) diffeomor- phism of the underlying smooth manifolds such that ϕ∗ : (TxM, fx)→ ( Tϕ(x)M ′, f′ϕ(x) ) is an iso- morphism of filtered vector spaces for all x ∈M , where ϕ is defined. More generally, we can define a morphism of two filtered manifolds ϕ : (M, f) → (M ′, f′) as an arbitrary smooth map ϕ : M → M ′ such that ϕ∗ : TxM → Tϕ(x)M ′ preserves the filtrations, that is ϕ∗(f p x) ⊂ f′ pϕ(x). Let (M, f) be a filtered manifold. For x ∈M we set grp fx = fpx/f p+1 x and gr fx = ⊕ p∈Z grp fx. We see easily that gr fx inherits a Lie bracket induced from the bracket of vector fields, which satisfies[ grp fx, grq fx ] ⊂ grp+q fx for all p, q ∈ Z. Hence, gr fx turns to be a nilpotent graded Lie algebra, which is called the symbol algebra of (M, f) at x. It should be noted that gr fx and gr fx′ are not necessarily isomorphic as graded Lie algebras for different x, x′ ∈M . If, for all x ∈M , the Lie algebra gr fx is isomorphic to a certain fixed graded Lie algebra m = ⊕p∈Zmp, we say that (M, f) is of type m. Lemma 2.1. Let ϕ : (M, f)→ (M ′, f′) be an immersion of two filtered manifolds. Then for each x ∈M the map ϕ∗ induces a homomorphism gr fx → gr f′ϕ(x) of graded Lie algebras. In particu- lar, if ϕ is a local diffeomorphism, then the Lie algebras gr fx and gr f′ϕ(x) are isomorphic. Proof. Indeed, consider arbitrary u ∈ grp fx, v ∈ grq fx. Let U ∈ fp, V ∈ fq such that u = Ux mod fp+1 and v = Vx mod fq+1. Then ϕ∗(U), ϕ∗(V ) are well-defined vector fields on ϕ(M) in some neighborhood of ϕ(x), and, moreover, ϕ∗([U, V ]) = [ϕ∗(U), ϕ∗(V )]. Locally we can extend ϕ∗(U), ϕ∗(V ) to vector fields U ′ ∈ (f′)p, V ′ ∈ (f′)q. Then we have [U ′, V ′]\|ϕ(M) = ϕ∗([U, V ]). As the definition of the bracket operations on gr f and gr f′ does not depend on the choice of representatives, we see that ϕ∗ indeed induces the Lie algebra homomorphisms gr fx → gr f′ϕ(x) for all x ∈M . � Remark 2.2. Note that the induced homomorphism of graded Lie algebras is not injective in general, even if ϕ∗ is injective at each point. Example 2.3 (trivial filtered manifold). Any smooth manifoldM can be considered as a (trivial) filtered manifold (M, f) with f0 = 0 and f−1 = TM . It is clear that gr fx is a commutative Lie algebra of dimension dimM for all x ∈ M . Hence gr fx is identified with TxM regarded as a commutative Lie algebra. Example 2.4 (filtered manifold generated by a differential system D). Let M be an arbitrary smooth manifold, and let D ⊂ TM be a vector distribution. If f−1 = D, f−k−1 = f−k + [ f−1, f−k ] ∀k > 0, we say that f = {fp} is generated by D = f−1, or alternatively the vector distribution D is (regularly) bracket generating. The symbol algebra gr fx is also called the symbol of the distribution D at a point x ∈ M . Note that in this case gr fx is necessarily generated by gr−1 fx. Extrinsic Geometry and Linear Differential Equations 7 Example 2.5 (homogeneous filtered manifold). Let G/G0 be a homogeneous space with a Lie group G and its closed subgroup G0. Assume that the Lie algebra g of G is endowed with a filtration {gp} such that (i) [gp, gq] ⊂ gp+q, (ii) Ad(a)gp ⊂ gp for a ∈ G0 (p < 0), (iii) g0 is the Lie algebra of G0. Then the filtration {gp} induces a left G-invariant tangential filtration fG/G0 on G/G0. We see immediately that gr(fG/G0)x ∼= gr− g ( = ⊕ p<0 grp g ) for any x ∈ G/G0. In particular, if m = ⊕ p<0 mp is a nilpotent graded Lie algebra, the simply connected Lie group M corresponding to m endowed with the left invariant tangential filtration is called the standard filtered manifold of type m. 2.2 Flag varieties Let V be a vector space over R or C, and be finite dimensional unless otherwise stated. By a fil- tration φ of V we mean a series {φp}p∈Z of subspaces φp of V satisfying φp ⊃ φp+1, that is, our filtrations are descending. A filtration φ is called saturated if ∪φp = V , and fine if ∩φp = 0. Unless otherwise stated, we shall always assume that all filtrations are saturated and fine. Two filtrations φ1, φ2 are said to be isomorphic if there is a linear isomorphism f of V such that fφp1 = φp2 for all p. Let Flag(V ) denote the flag space of all filtrations of V , and Flag(V, φ) the flag variety of type φ, that is, the set of all filtrations of V isomorphic to φ. Note that the filtration φ of V induces that of gl(V ), denoted by the same letter φ: φpgl(V ) = { A ∈ gl(V ) : AφiV ⊂ φi+pV } , which satisfies[ φpgl(V ), φqgl(V ) ] ⊂ φp+qgl(V ). Denote also by GL(V )0 the subgroup of GL(V ) stabilizing the flag φ. Its Lie algebra is exactly φ0gl(V ). Note that the flag variety Flag(V, φ) can be naturally identified with the homogeneous space GL(V )/GL(V )0. Since the filtration {φpgl(V )} is invariant with the adjoint action of GL(V )0, from Example 3 we see that Flag(V, φ) is a homogeneous filtered manifold. 2.3 Osculating maps Let (M, f) be a filtered manifold. A smooth map ϕ : (M, f)→ Flag(V, φ) is called osculating if fp ϕq ⊂ ϕp+q for all p, q ∈ Z, where ϕq denotes the vector bundle ∪x∈Mϕq(x) over M induced from ϕ and ϕq the sheaf of sections of ϕq, and the multiplication on the left hand side of the above formula signifies the operation by differentiation along vector fields by viewing ϕq ⊂ V ×M . It is easy to see that this condition is equivalent to the fact that ϕ is a morphism of filtered manifolds. Let ϕ : (M, f)→ Flag(V, φ) be an osculating map. For x ∈ M , denote by grϕ(x) the graded vector space ⊕ ϕp(x)/ϕp+1(x). Then we have the associated map grp fx × grq ϕ(x)→ grp+q ϕ(x). By this operation grϕ(x) becomes a gr fx-module. We say that an osculating map ϕ : (M, f) → Flag(V, φ) is generated by {ϕp}p≥p0 if grϕ(x) is generated by ⊕ p≥p0 grp ϕ(x) for all x ∈ M . In other words, this means that the map ϕ is completely determined by ϕp0 as follows ϕp = ∑ i>0 f−i ϕp+i for all p < p0. 8 B. Doubrov, Y. Machida and T. Morimoto Actually, any map from a filtered manifold to a projective space or to a Grassmann variety generically generates an osculating map to a flag variety. We say that an osculating map ϕ : (M, f)→ Flag(V, φ) has constant symbol of type (g−, grV ) if there exist a nilpotent graded Lie algebra g− = ⊕ p<0 gp and a graded g−-module structure on gr(V, φ) such that (gr fx, grϕ(x)) is isomorphic to (g−, grV ) as graded modules for all x ∈M , that is, there exist an graded linear isomorphism α : grV → grϕ(x) and a graded Lie algebra isomorphism β : g− → gr fx which make the following diagram commutative: g− × grV −−−−→ grV β×α y yα gr fx × grϕ(x) −−−−→ grϕ(x). 3 Three categories In this section we define three categories associated to L/L0 ⊂ Flag(V, φ), where (V, φ) is a fil- tered vector space, L is a Lie subgroup of GL(V ) and L0 the subgroup of L which fixes φ. We denote the induced filtration of gl(V ) by {φpgl(V )} or simply by {gl(V )p} and define the filtration {lp} of the Lie algebra l of L by lp = l ∩ gl(V )p. Similarly we set Lp = L ∩ GL(V )p for p ≥ 0. We denote by gr l the associated graded Lie algebra and gr− l = ⊕ p<0 grp l. 3.1 Category of L/L0 extrinsic geometries Definition 3.1. Every object of the category of L/L0 extrinsic geometries is the osculating map ϕ : (M, f)→ L/L0 ⊂ Flag(V, φ). Every morphism is a pair (f,Λa) of maps defined by the following commutative diagram: (M, f) ϕ−−−−→ L/L0 ⊂ Flag(V, φ)yf yΛa (M ′, f′) ϕ′−−−−→ L/L0 ⊂ Flag(V, φ), where f is a diffeomorphism of filtered manifolds and Λa is a left translation by a ∈ L. Definition 3.2. An L/L0 extrinsic geometry is called of type (g−, grV,L) at x ∈ M , if there exist a graded Lie subalgebra g− ⊂ gr− l, a graded Lie algebra isomorphism β : g− → gr fx and a ∈ L satisfying aφ = ϕx which make the following diagram commutative: g− × grV −−−−→ grVyβ×α yα gr fx × grϕ(x) −−−−→ grϕ(x), where α = gr a. It is called (of constant symbol) of type (g−, grV,L), if it is so for all x ∈M . 3.2 Category of L/L0 differential equations Definition 3.3. Every object of the category of L/L0 (weighted) involutive systems of linear differential equations (or simply the category of L/L0 differential equations) is an L0-filtered Extrinsic Geometry and Linear Differential Equations 9 vector bundle (R, {Rq}) with a typical fibre (V, {V q}) over a filtered manifold (M, f) equipped with a flat l-connection ∇ on R satisfying ∇fpR q ⊂ Rp+q for all p, q. Every morphism of this category is an isomorphism ( R, {Rq}, (M, f),∇ ) → ( R′, {R′q}, (M ′, f),∇′ ) of L0-filtered vector bundles, filtered manifolds and l-connections. We mean by an L0-filtered vector bundle with a typical fibre ( V, {V q} ) a vector bundle whose structure group is specified to be L0 ⊂ φ0GL(V ), namely a vector bundle endowed with a maximal class of admissible local trivializations M × V covering the bundle such that the transition function of any two admissible trivializations takes values in L0. If π : R→M is an L0-vector bundle with a typical fibre V and if U × V 3 (x, v) 7→ ζ(x)v ∈ π−1(U) is an admissible local trivialization, we call ζ or ζ(x) an admissible frame of R. If we fix a basis {e1, e2, . . . , em} of V and set ζi(x) = ζ(x)ei, then {ζ1(x), ζ2(x), . . . , ζm(x)} is an admissible basis of the frame Rx. Recall that a connection on a vector bundle E →M is a 1-st order linear differential operator ∇ : E → E ⊗ T ∗M satisfying ∇(fσ) = f∇σ + σ ⊗ df for σ ∈ E, f ∈M × R. Now as in Definition 3.3, let R be an L0-vector bundle and ∇ a connection on R in the usual sense defined above, and let L0 ⊂ L ⊂ GL(V ). We say that ∇ is an l-connection if ζ−1∇Xζ ∈ l for any L0-admissible frame ζ of R and X ∈ TM , where l is a Lie algebra of L. We say that the connection ∇ is flat if its curvature K vanishes identically, where K(X,Y ) = ∇X∇Y −∇Y∇X −∇[X,Y ]. Let ( (R, {Rp}), (M, f),∇ ) be an L/L0 differential equation. For each x ∈M , we have gr fx × grRx → grRx, which makes grRx a graded gr fx-module called the symbol of R at x. Definition 3.4. An L/L0 differential equation ( R, {Rq},∇ ) over (M, f) is of type (g−, grV,L) at x ∈ M , if there exist a graded Lie subalgebra g− ⊂ gr− l, an isomorphism β : g− → gr fx of graded Lie algebras and z ∈ Πx such that the following diagram is commutative: g− × grV −−−−→ grVyβ×gr z ygr z gr fx × grRx −−−−→ grRx. It is called (of constant symbol) of type (g−, grV,L) if it is so for all x ∈M . 10 B. Doubrov, Y. Machida and T. Morimoto 3.3 Category of L/L0 extrinsic bundles Let K0 be an arbitrary subgroup of L0, and let k0 ⊂ l0 be the corresponding subalgebra. Note that it naturally inherits the filtration from l: kp = lp ∩ k0. Definition 3.5. Every object in the category of L/L0 extrinsic bundles is a principal K0-bundle π : P → M over a filtered manifold (M, f) endowed with an l-valued 1-form ωP (or simply written ω) such that (i) R∗aω = Ad(a)−1ω, where Ra, a ∈ K0, denotes the right shift on the principal bundle, (ii) 〈Ã, ω〉 = A for any fundamental vector field Ã generated by A ∈ k0, (iii) dω + 1 2 [ω, ω] = 0. Every morphism of this category is an isomorphism of L/L0 extrinsic bundles, that is, a bundle isomorphism preserving l-valued 1-forms. It should be remarked that if (P,K0, ω) is an L/L0 extrinsic bundle over (M, f), then the image of gr ω̄z : gr fx → gr− l is a graded Lie subalgebra of gr− l. Indeed, let u ∈ grp fx and v ∈ grq fx. Take X ∈ fpx, Y ∈ fqx such that Xx ≡ u, Yx ≡ v, and let X̃, Ỹ be lifts of X, Y to P respectively. Then[ gr ω̄z(u), gr ω̄z(v) ] ≡ [ ωz ( X̃ ) , ωz ( Ỹ )] = −dω ([ X̃, Ỹ ]) = −X̃ω ( Ỹ ) + Ỹ ω ( X̃ ) + ω ([ X̃, Ỹ ]) ≡ ωz ([ X̃, Ỹ ]) ≡ ωz ( [̃X,Y ] ) ≡ gr ω̄z ( [u, v] ) . We say that ( P,K0, ω ) is (of constant symbol) of type ( g−, L/L 0 ) if for any x ∈ M there exists z ∈ π−1(x) such that gr ω̄z(gr fx) = g− for a graded Lie subalgebra g− ⊂ gr− l. 3.4 Congruence classes Let us define the congruence classes in each three categories in order to state rigorously the functorial isomorphisms that we shall prove in the next section. We say two L/L0 extrinsic geometries ϕ1 : (M1, f1) → L/L0 and ϕ2 : (M2, f2) → L/L0 are congruent by a congruence (idM ,Λa) if M1 = M2 = M and if there exists an isomorphism (idM ,Λa) from ϕ1 to ϕ2 of L/L0 extrinsic geometries. We say also that two isomorphisms of extrinsic geometries (f,Λa) : [ ϕ1 : (M1, f1)→ L/L0 ] → [ ϕ2 : (M2, f2)→ L/L0 ] , (f ′,Λa′) : [ ϕ1 : (M1, f1)→ L/L0 ] → [ ϕ2 : (M2, f2)→ L/L0 ] are congruent if there exist congruences (idM1 ,Λb1) and (idM2 ,Λb2) such that (idM2 ,Λb2) ◦ (f,Λa) = ( f ′,Λa′) ◦ (idM1 ,Λb1 ) . Then we have a category whose objects are the congruent classes of extrinsic geometries and the morphisms the congruence classes of isomorphisms. In accordance with the above definition, we use the terminology: “congruence” and “cong- ruent” also for (vector or fibre) bundles to mean by a congruence an isomorphism of bundles which covers the identity map of the base spaces. Then we have the category of congruence classes of L/L0 differential equations and that of congruence classes of L/L0 extrinsic bundles. Extrinsic Geometry and Linear Differential Equations 11 But for L/L0 extrinsic bundles we introduce another notion of congruence in a wider sense, called holo-congruent. Before that let us make some observation on a certain generalization of connection. Let G be a Lie group and K a Lie subgroup with Lie algebra g and k respectively. Let Q → M be a principal K-bundle over M . A g-valued 1-form ωQ (or simply ω) on Q will be called a g-connection (form) if it satisfies: (i) R∗aω = Ad(a)−1ω for a ∈ K, (ii) 〈ω, Ã〉 = A for A ∈ k. Cartan connections in intrinsic geometry and L/L0 extrinsic bundles are examples of g- connections. If ω is a g-connection on Q and if we set dω + 1 2 [ω, ω] = γ, then iÃγ = 0, R∗aγ = Ad(a)−1γ. Therefore γ is be viewed as a map ∧2(TQ/V Q)→ g, where V Q denotes the vertical tangent bundle of Q, and is called the curvature of ω. If γ = 0, we say that ω is flat. Proposition 3.6. Let G ⊃ H ⊃ K ⊃ L be a descending sequence of Lie subgroups and Q→M a principal K-bundle equipped with a g-connection ωQ. Then (1) If R is principal L-bundle over M and ι : R → Q is an injection of principal bundles covering idM , then ωR(= ι∗ωQ) is a g-connection on R, and ωQ is flat if and only if so is ωR. (2) There exist, uniquely up to congruences, a principal H-bundle P → M , a g-connection on ωP on P , and an injection ι : Q → P of principal bundles covering idM such that ι∗ωP = ωQ. Proof. Let us indicate how to construct (P, ωP ). First we define P to be the fibre product Q×K H. We then define a g-valued 1-form ωQ×H on Q×H by (ωQ×H)(q,h) = Ad(h)−1(ωQ)q + (ΩH)h for (q, h) ∈ Q×H, where ΩH denotes the Maurer–Cartan form of H. It is not difficult to verify that there exists a unique g-valued 1 form ωP on P such that its pull-back to Q×H coincides with ωQ×H . Then it is easy to prove the other statements of the proposition. We remark that a simple computation yields dωQ×H + 1 2 [ωQ×H , ωQ×H ] = Ad(h)−1 ( dωQ + 1 2 [ωQ, ωQ] ) , which tells how to compute the curvature by taking a local trivialization of Q and also gives an alternative proof of the second half of (1). � 12 B. Doubrov, Y. Machida and T. Morimoto Now let us return to the category of L/L0 extrinsic bundles. By the above proposition we see that for an L/L0 extrinsic bundle ( Q,K0,M, ωQ ) there is, uniquely up to congruences, an extension of Q to the group L0 which we denote ( Q̄, L0,M, ωQ̄ ) . We say that two L/L0 extrinsic bundles Q and Q′ are holo-isomorphic (resp. holo-congruent) if their extensions to L0, Q̄ and Q̄′ are isomorphic (resp. congruent). Then the holo-conguruence classes of L/L0 extrinsic bundles forms a category, of which the morphisms are the congruence classes of holo-isomorhisms. 4 Categorical isomorphisms In this section we show that the categories of congruence classes of L/L0 extrinsic geometries, congruence classes of L/L0 differential equations, and holo-congruence classes of L/L0 extrinsic bundles are categorically equivalent. The passage to the L/L0 extrinsic geometries needs the topological assumption of simply connectedness. Once recognized we will not be strict for the distinction between X and the (holo-) congruence class of X, if there is no fear of confusion. 4.1 From an extrinsic geometry to an extrinsic bundle Let ϕ : (M, f) → L/L0 ⊂ Flag(V, φ) be an L/L0 extrinsic geometry. Let ϕ∗L be the induced principal fibre bundle over M . Then its structure group is L0 and there is a canonical embedding ι : ϕ∗L → L. Let ω be the pull back ι∗ωL, where ωL is the Maurer–Cartan form of L. Then (ϕ∗L,L0, ω) is an L/L0 extrinsic bundle over (M, f). 4.2 From an extrinsic bundle to an extrinsic geometry Let ( P,K0, ωP ) be an extrinsic bundle. We then consider the Pfaff system on P ×L defined by Ω = ωL − ωP = 0. (Σ) Since we have dΩ = −1 2 [Ω, ωL + ωP ] , the Pfaff system (Σ) is completely integrable. Therefore for each (z, a) ∈ P × L there is a unique maximal integral manifold, which defines a fibre preserving map from a neighborhood of z to a neighbourhood of a. It then defines an embedding of a neighborhood of the projection π(z) ∈ M to L/L0. Note that any such two embeddings differ by a left translation of an element of L. Therefore if M is simply connected, the maximal integral manifold passing (z0, a0) determines an osculating map ϕ : M → L/L0, which gives the functorial isomorphism. 4.3 From a differential equation to an extrinsic bundle Let ( R, {Rp}, (M, f),∇ ) be an L/L0 differential equation. Then define P to be the set of admis- sible frames of the L0-vector bundle R. We see that P is an L0-principal fibre bundle over M , and R can be identified with P ×L0 V . Let Π be the extension of P to L: Π = P ×L0 L. Then the l-connection ∇ of R gives rise to a connection on Π in the following way. A connection on Π is usually defined by a right invariant distribution H of horizontal sub- spaces, or by a connection form ω, which is an l-valued 1-form on Π satisfying (i) R∗aω = Ad(a)−1ω, for all a ∈ L, (ii) 〈ω, Ã〉 = 0, for all A ∈ l. Extrinsic Geometry and Linear Differential Equations 13 The connection H associated to ∇ is defined as follows. Let z̊ ∈ Π and x̊ ∈M its projection. For any smooth curve ξ(t) ∈ M with ξ(0) = x̊ there is a locally unique horizontal lift ζ(v)(t) to R such that ζ(v)(0) = z̊(v) for v ∈ V . Then ζ(t) is a lift of ξ(t) to Π. Tangent vectors to all such lifts define a horizontal subspace Hz̊ ⊂ Tz̊Π. The connection form ω is uniquely defined by kerω = H. Then the following properties are standard: Proposition 4.1. (1) ∇ζ = ζ(ζ∗ω) for any local section ζ of Π, (2) for local sections ζ and σ of Π and R respectively if we define η̄ by σ = ζη̄, then ∇σ = ζ (dη̄ + ω̄η̄), where ω̄ = ζ∗ω, (3) iÃ ( dω + 1 2 [ω, ω] ) = 0 for all A ∈ l, (4) K = dω + 1 2 [ω, ω]. Proof. Since (1) and (2) are basic, we give a short proof. Let̊x ∈ M , z̊ = ζ (̊x), ξ ∈ Tx̊M and x(t) be a smooth curve in M such that x(0) = x̊, ẋ(0) = ξ. Let zH(t) be its horizontal lift satisfying zH(0) = z̊. Then writing ζ(t) = ζ(x(t)) = zH(t)a(t) with a(t) ∈ L, a(0) = e, we have ζ−1∇ξζ = ζ−1∇ξ(zHa) = ζ−1 ( (∇ξzH)a(0) + ζ(0)a(0)−1ȧ(0) ) = ȧ(0), 〈ζ∗ω, ξ〉 = 〈ω, ζ̇(0)〉 = 〈 ω, żH(0)a(0) + d dt \|t=0 ( zH(0)a(t) )〉 = ȧ(0). Hence (1) is verified. Now (2) immediately follows from (1): ∇σ = ∇(ζη̄) = (∇ζ)η̄ + ζdη̄ = ζ(dη̄ + ω̄η̄). Note that on account of (3) K = dω+ 1 2 [ω, ω] may be regarded as a section of Hom ( ∧2TM, l ) . Then (4) follows easily from (2). � On the bundle π : Π→ (M, f) there is an induced filtration fpΠ defined by fpΠ = π∗fp, p < 0, fpΠ = (̃φpl), p ≥ 0. The properties of Proposition 4.1 imply Proposition 4.2. Let ∇ be a flat l-connection on R and let (Π, ω) be the associated principal L-bundle and connection. The following two conditions are equivalent: (1) ∇fpR q ⊂ Rp+q for all p, q, (2) ωz : TzΠ→ l is filtration preserving for any z ∈ Π. If ι : P → Π is the canonical inclusion, then it is easy to see that (P, ι∗ω) in an L/L0-extrinsic bundle. 14 B. Doubrov, Y. Machida and T. Morimoto 4.4 From an extrinsic bundle to a differential equation Let ( P,K0, ω ) be an L/L0 extrinsic bundle over a filtered manifold (M, f). Then define a filtered vector bundle (R, {Rq}) by Rq = P ×K0 V q, which can be regarded as L0-filtered vector bundle by group extension. The l-valued 1-form ω then defines an l-connection on R. It is now clear that (R, {Rq},∇) is an L/L0-differential equation. It should be noted that the L/L0 differential equation corresponding to the L/L0 extrinsic bundle ( P,K0, ω ) is written succinctly as dη + ω · η = 0. ( Σ) Here by · we mean the action of the Lie algebra l on V . This is a first order linear differential equation for unknown V -valued function η on P . Since ω is flat, this equation is integrable. In fact, if we view ( Σ) as a Pfaff equation on P × V by regarding η as the standard coordinates of V and put Θ = dη + ω · η, then we have dΘ = dω · η − ω ∧ dη = −1 2 [ω, ω] · η − ω ∧ (Θ− ω · η) = −ω ∧Θ. Hence ( Σ) is completely integrable. Therefore for any (̊z, v̊) ∈ P×V there exists locally a unique solution η of ( Σ) such that η(̊z) = v̊. Moreover, since Ãη = −A · η for A ∈ k0, we have η ( z exp(tA) ) = ( exp(tA)−1 ) η(z). Therefore η can be regarded as a local section σ of the associated bundle R = P ×K0 V . Then we see immediately from Proposition 4.1(2) that a solution η of ( Σ) corresponds bijectively to a section σ of R satisfying ∇σ = 0. 4.5 From an extrinsic geometry to a differential equation Let ϕ : (M, f) → L/L0 ⊂ Flag(V, φ) be an L/L0 extrinsic geometry. Then, taking Rq := ∪x∈Mϕq(x) (ϕq(x) ⊂ V ), R := M × V and ∇fpR q := fpRq, that is, ∇ = d for V -valued functions on M , we have an L/L0 differential equation ( R, {Rp},∇ ) on (M, f). 4.6 From a differential equation to an extrinsic geometry Let ( R, {Rp},∇ ) be an L/L0 differential equation on (M, f). One way to construct the corre- sponding L/L0 extrinsic geometry is to take an extrinsic bundle ( P,K0, ωP ) and integrate ωP to obtain a map g : P → L. Here we like to give another construction. We assume the base space M is simply connected. Let Sol(∇) denote the space of all solutions of ∇s = 0. Since there exists a unique solution s such that s(x0) = v0 for arbitrary given x0 ∈M, v0 ∈ Rx0 , we note that Sol(∇) ∼= Rx0 by evaluation and Rx0 ∼= V by any z0 ∈ Π over x0. Extrinsic Geometry and Linear Differential Equations 15 Now define a map ϕ : (M, f)→ Flag(Sol(∇)) by ϕp(x) = { s ∈ Sol(∇) : s(x) ∈ Rpx } . Then this map ϕ is exactly the corresponding extrinsic geometry. To say in the language of extrinsic bundles, let Sol( Σ) denote the solution space of ( Σ), which is isomorphic to V by an evaluation Sol( Σ) 3 f 7→ f(z0) ∈ V z0 ∈ P. Now we define Ψ: P → Flag(Sol( Σ)) by Ψp(z) = { f ∈ Sol( Σ): f(z) ∈ V q } , which in turn induces a map ψ : (M, f)→ Flag(Sol( Σ)). Then we see easily that ϕ and ψ are congruent, if we regard them as maps to Flag(V, φ) by any admissible isomorphisms α : Sol(∇)→ V , β : Sol( Σ)→ V specified above. We remark here that, if we obtain a map g : P → L such that g∗ωL = ωP , then we have immediately Sol( Σ) = { g−1v : v ∈ V } , which implies that integration of ωP solves ( Σ). 4.7 L/L0 differential equations as linear differential equations in weighted jet bundles Let us show that an L/L0 differential equation (R, {Rp},∇) can be realized as an involutive system of linear differential equations of finite type defined in a weighted jet bundle (see [23] for weighted jet bundles). Let R(ν) denote the quotient bundle R/Rν+1, which is a filtered vector bundle over the filtered manifold (M, f), and let ĴkR(ν) be the weighted jet bundle of weighted jet order k ≥ 0 associated to R(ν). Proposition 4.3. There are canonical morphisms j : R(k) → ĴkR(ν), for any k, ν, therefore R(k) may be regarded as a system of linear differential equations in ĴkR(ν). Proof. The map j is defined as follows. Let v(k) ∈ R(k) x . Take a section v ∈ Rx such that ∇v = 0 and v(k)(x) = v(k), 16 B. Doubrov, Y. Machida and T. Morimoto where v(k) denotes the projection of v into R(k). We then define j ( v(k) ) = jkx ( v(ν) ) , where jkx denotes the weighted k-th jet at x. Let us show that this definition does not depend on the choice of v and is well-defined. For that we are going to verify that if v(k) = 0, then j (k) x v(ν) = 0. Take a local cross-section ζ of Π in a neighborhood of x. Then to the section, v corresponds a V -valued function u in a neighborhood of x determined by v(y) = ( ζ(y), u(y) ) . Taking complementary subspaces Vp to V p+1 in V p, we can write V = ⊕Vp and u = ∑ up with up a Vp-valued function. Now recall that the coordinates of jkxv (ν) are represented by( Xp1 · · ·Xpτuj ) x for Xpi ∈ fpi , j ≤ ν with j − ∑ pi ≤ k. We claim that all these values vanish, if v(k) = 0. Let us verify it in a special case τ = 2, the other cases being similar. Since ∇v = 0, we have, by Proposition 4.1(2), Xu+ 〈ζ∗ω,X〉u = 0. If we write ω̃ = −ζ∗ω, we have Xu = ω̃(X)u = ∑ ω̃ji (X)uj , where ω̃ji (X) denotes Hom(Vj , Vi) component. Now we have Xp1Xp2u = Xp1 ( ω̃(Xp2)u ) = ( Xp1ω̃(Xp2) ) u+ ω(Xp2)Xp1u = ( Xp1ω̃(Xp2) ) u+ ω̃(Xp2)ω̃(Xp1)u. Hence we have Xp1Xp2ui = ∑ j ( Xp1ω̃ j i (Xp2) ) uj + ∑ j,l ω̃ji (Xp2)ω̃lj(Xp1)ul. Note here that by Proposition 4.2 we have ω̃ji (Xp) = 0 for Xp ∈ fp, if i < j + p. Therefore Xp1Xp2ui = ∑ j≤i−p2 ( Xp1ω̃ j i (Xp2) ) uj + ∑ j≤i−p2 ω̃ji (Xp2) ∑ l≤j−p1 ω̃lj(Xp1)ul. But the values uj(x) for j ≤ i− p2 ≤ k, ul(x) for l ≤ j − p1 ≤ i− p1 − p2 ≤ k vanish. Hence (Xp1Xp2ui)x for i− p1 − p2 ≤ k vanish, which completes the proof of Proposition 4.3. � Extrinsic Geometry and Linear Differential Equations 17 It should be remarked that the morphism j is actually defined without “integration”. Though we have used integration to find v such that ∇v = 0, it is not v, but v(k) that plays the actual role to define j as shown in the proof. Let Hk(gr f, grR) be the degree k cohomology group of the gr f module grR. As both gr f and grR are graded, the cohomology group also naturally inherits the grading Hk(gr f, grR) = ∑ i Hk i (gr f, grR). Proposition 4.4. If H0 i (gr f, grR) = 0 for i > ν, then the map j : R(k) → ĴkR(ν) is injective for k ≥ ν. Proof. Consider the following commutative diagram (see [23]): 0 0y y grk+1R −−−−→ Hom(U(gr f), grR(ν))k+1y y R(k+1) −−−−→ Ĵk+1R(ν)y y R(k) −−−−→ ĴkR(ν)y y 0 0 The third row is injective for k = ν and the first row is injective for k ≥ ν by assumption. Then by induction we deduce that the second row is injective for k ≥ ν, since the columns are exact. � Proposition 4.5. Let k1 and k2 be integers such that ν ≤ k1 ≤ k2, H1 j (gr f, grR) = 0 for j > k1 and grlR = 0 for l > k2. For k ≥ ν, consider R(k) as systems of differential equations embedded in ĴkR(ν). Then they satisfy (1) Prol(l)R(k) ⊂ R(l) for l ≥ k ≥ ν, (2) Prol(l)R(k) = R(l) for l ≥ k ≥ k1, (3) R→ · · · → R(k+1) → R(k) are all isomorphisms for k ≥ k2 and are involutive. Proof. For the subbundle R(k) ⊂ ĴkR(ν) the prolongation Prol(l)R(k) ⊂ Ĵ lR(ν) is defined for l ≥ k (for the definition see [23]). All assertions in Proposition 4.5 can be easily verified by standard arguments in the theory of weighted jet bundles. � Existence of such integers ν, k1, k2 is theoretically obvious in our finite-dimensional case. Such minimal integers are of interest for various concrete examples. 4.8 Dual embeddings and differential equations Let V be a vector space and V ∗ its dual space. For a decreasing filtration φ = {φp} we define the dual of φ to be the filtration φ∗ of V ∗ defined by φ∗p = ( φ−p+1 )⊥ . 18 B. Doubrov, Y. Machida and T. Morimoto Then the pairing grp φ× grq φ ∗ → R is non-degenerate if p+ q = 0 and vanishes otherwise. Therefore (grφ)∗ ∼= grφ∗. For a map ϕ : (M, f) → Flag(V, φ) we define its dual ϕ∗ : (M, f) → Flag(V ∗, φ∗) by ϕ∗(x) = ϕ(x)∗ for x ∈M . Proposition 4.6. 1. The map ϕ is osculating if and only if so is ϕ∗. Assume that ϕ is osculating. Then 2. (grϕ(x))∗ ∼= grϕ∗(x) and 〈Xα, v〉+ 〈α,Xv〉 = 0 for X ∈ gr fx, α ∈ grϕ∗(x), v ∈ grϕ(x). 3. H0 i (gr fx, grϕ(x)) = 0 for i > λ if and only if grϕ∗(x) is generated by ⊕j≥−λ grj ϕ ∗(x). The proof is straightforward and is omitted. It should be remarked that if ϕ : (M, f) → L/L0 ⊂ Flag(V, φ) is a L/L0-extrinsic geome- try, then ϕ∗ is also an L/L0-extrinsic geometry with respect to the dual embedding L/L0 ⊂ Flag(V ∗, φ∗). We also note that the duality in L/L0 extrinsic geometries naturally extends to that in L/L0 differential equations. 4.9 Examples In the examples below we illustrate the relation between differential equations, the corresponding osculating embeddings and their dual embeddings. Throughout these examples we denote by V (or V k+1) a vector space of dimension k+1, and by V ∗ its dual. By taking a basis {e0, e1, . . . , ek} of V and its dual basis {e0, e1, . . . , ek} of V ∗ we make often the identifications V 3 v = ∑ viei ↔  v0 v1 ... vk  ∈ Rk+1, V ∗ 3 α = ∑ αie i ↔ (α0, α1, . . . , αk) ∈ Rk+1,∗. We write M or Mn to denote a filtered manifold of dimension n endowed with the trivial filtration unless otherwise mentioned. Example 4.7. Consider a curve [θ] : M1 → P ( V k+1,∗) represented by a smooth curve θ : M1 → V k+1,∗\{0}. Assume that [θ] is regularly generating V ∗, that is there exists an osculating map ϕ∗ : M → Flag(V ∗, φ∗) such that ϕ∗0(x) = 〈θ(x)〉 ⊃ ϕ∗1(x) = 0 for all x ∈ M and that ϕ∗ is generated by ϕ∗0. Note that ϕ∗ is uniquely determined by [θ] up to a shift of filtration. In this case this assumption is equivalent to assuming that { θ(x), θ′(x), . . . , θ(k)(x) } are linearly independent and therefore span V ∗ for every x ∈M . Extrinsic Geometry and Linear Differential Equations 19 Then there exist smooth functions p0, p1, . . . , pk on M such that θ(k+1)(x) = p0(x)θ(k)(x) + · · ·+ pk(x)θ(x), for all x ∈M. This means that if we write θ = (θ0, θ1, . . . , θk) then {θ0, θ1, . . . , θk} forms a fundamental system of solutions of the following linear ordinary differential equation (D) of order k + 1: y(k+1) = p0y (k) + p1y (k−1) + · · ·+ pky. (D) Let ϕ : M → Flag(V, φ) be the dual of ϕ∗, i.e., ϕ = (ϕ∗)∗. Then by Proposition 4.6 the map ϕ is osculating and H0 +(gr fx, grϕ(x)) = 0. Hence, viewing L = GL(V ), L0 = φ0GL(V ), we see that the L/L0-differential equation R = ( {Rq},∇ ) corresponding to ϕ is defined in the jet bundle Jk+1(R(0)) by Proposition 4.5. Let us observe that the differential equation R is nothing but the ODE (D). Indeed, define ψ : M → Flag(Sol(D)) by Sol(D) = sv = k∑ i=0 viθi ∣∣∣∣∣∣ v = v0 ... vk  ∈ Rk+1  , ψp(x) = { sv ∈ Sol(D) \| s(i) v (x) = 0, that is 〈 θ(i), v 〉 = 0 for all i < p } . Clearly ψ is isomorphic to ϕ. But the differential equation corresponding to ψ is (D), so that the systems (D) and R are equivalent. On the other hand, we know that the differential equation R is equivalent to the Pfaff equation dη + ωη = 0, ( Σ) where ω is the pull-back of the Maurer–Cartan form ΩGL(V ) to the induced bundle ϕ∗GL(V ): ϕ∗GL(V ) −−−−→ GL(V )y y M −−−−→ GL(V )/φ0GL(V ). Let us see how the equation ( Σ) looks like in our case. Let {Θ0,Θ1, . . . ,Θk} be a moving frame of V ∗ defined by Θp = θ(k−p), p = 0, 1, . . . , k and let{ Φ0,Φ1, . . . ,Φk } be the moving frame on of V dual to {Θ0,Θ1, . . . ,Θk}. Regarding Θq as a row vector and Φp as a column vector, we set Θ =  Θ0 Θ1 ... Θk , Φ = ( Φ0,Φ1, . . . ,Φk ) , and we get GL(V )-valued functions Θ, Φ on M satisfying ΘΦ = Ek+1 (the identity matrix of degree k + 1). It is clear that Φ gives a cross-section of the bundle ϕ∗GL(V ) → M . Thus, the equation ( Σ) reduces to dη + Φ−1 dΦ η = 0, 20 B. Doubrov, Y. Machida and T. Morimoto which then becomes dη − dΘ Θ−1 η = 0. But by the construction of Θ we see that dΘ Θ−1 =  p0 p1 p2 . . . pk−1 pk 1 0 0 . . . 0 0 0 1 0 . . . 0 0 ... ... . . . . . . ... ... 0 0 0 . . . 0 0 0 0 0 . . . 1 0  dx. Thus, the Pfaff equation ( Σ) reduces to  η′k η′k−1 ... η′0  =  p0 p1 p2 . . . pk−1 pk 1 0 0 . . . 0 0 0 1 0 . . . 0 0 ... ... . . . . . . ... ... 0 0 0 . . . 0 0 0 0 0 . . . 1 0   ηk ηk−1 ... η0 . This is exactly the system of ODEs of 1st order equivalent to (D). The above discussion gives our formulation of the well-known correspondence of the category of scalar linear differential equations of order k + 1 and the category of non-degenerate curves in P k explored by Wilczynski [37]. In the most symmetric case we arrive at a rational normal curve C ⊂ P ( Rk+1,∗) given by the (dual) Veronese embedding [θVero] : P ( R2 ) → P ( Rk+1,∗), [( z0 z1 )] 7→ [( −z1 )k , ( −z1 )k−1 z0, . . . , ( z0 )k] . In terms of representations, it is the GL(2,R)-orbit in P ( SkR2,∗) of the highest weight vector( ε1 )k ∈ SkR2,∗, where { ε0, ε1 } is the standard basis of R2,∗, and SkR2,∗ denotes the k-th sym- metric power of R2,∗, which has a basis {( ε1 )k , ( ε1 )k−1 ε0, . . . , ( ε0 )k} . The dual representation of GL(2,R) on R2,∗ extends naturally to P ( SkR2,∗). The stabilizer of [( ε1 )k] is B, the subgroup of upper-triangular matrices in GL(2,R). Thus we have the embedding P ( R2 ) = GL(2,R)/B → GL(2,R) · [( ε1 )k] , whose coordinate expression is the one given above. In terms of the affine coordinate x = z1/z0 the curve [θ] is given by θVero : R1 3 x 7→ ( (−x)k, (−x)k−1, . . . , (−x), 1 ) ∈ Rk+1,∗. The components of θ form a fundamental system of solutions of y(k+1) = 0. Thus, starting from the dual Veronese embedding [θ]Vero : P ( R2 ) → P ( Rk+1,∗), we have an osculating map ϕ∗Vero : P ( R2 ) → Flag ( Rk+1,∗, φ∗ ) generated by [θ]Vero and its dual ϕVero: P ( R2 ) → Flag ( Rk+1, φ ) . Then the differential equation corresponding to ϕVero is equivalent to the simplest one y(k+1) = 0. Extrinsic Geometry and Linear Differential Equations 21 Example 4.8. Consider a smooth surface in 3-dimensional projective space [θ] : M2 → P ( V 4,∗) represented by a smooth map θ : M2 → V 4,∗\{0}. Assume that [θ] generates an osculating map ϕ∗ : M → Flag(V ∗, φ∗) with ϕ0∗(x) = 〈θ(x)〉 ⊃ ϕ1∗(x) = 0, x ∈M . There are two cases to distinguish for the type of the map ϕ∗, that is when the sequence( dimV, . . . , dimφ−1, dimφ0 ) is equal to (4, 3, 2, 1) (case (i)) or to (4, 3, 1) (case (ii)). Taking local coordinates (x, y) on M and identifying V 4,∗ with R4,∗ write θ = ( θ0(x, y), . . . , θ3(x, y) ) . In case (i) we see that dim 〈 θ(p) 〉 = 1, dim 〈 θ(p), θx(p), θy(p) 〉 = 2, dim 〈 θ(p), θx(p), θy(p), θxx(p), θxy(p), θyy(p) 〉 = 3, dim 〈 θ(p), θx(p), . . . , θyyy(p) 〉 = 4. By a choice of coordinates we can assume that θy = 0, from which it follows that θxy = θyy = 0, so that θ, θx, θxx, θxxx are independent and the other derivatives of θ are expressed as their linear combinations. Thus, we have{ θy = 0, θxxxx = p0θxxx + p1θxx + p2θx + p3θ. Let ϕ be the dual of ϕ∗. Then by the same argument as in Example 4.7, we see that the differential equation corresponding to ϕ is equivalent to{ uy = 0, uxxxx = p0uxxx + p1uxx + p2ux + p3u, (Di) which reduces essentially to a linear ODE. Next, let us consider case (ii). For each point a ∈M grϕ∗(a) = gr−2 ϕ ∗(p) + gr−1 ϕ ∗(a) + gr0 ϕ ∗(a) is a gr fa-graded module. Here we have dim gr−2 ϕ ∗ = dim gr0 ϕ ∗ = 1, dim gr−1 ϕ ∗ = 2 and gr fa = TaM is a 2-dimensional abelian Lie algebra. If we take bases {ζ−2} and {ζ0} of gr−2 ϕ ∗ and gr0 ϕ ∗ respectively, then we have a symmetric bilinear form β on TaM defined by uvζ0 = β(u, v)ζ−2, which is equal to uvθ mod θ(a), θx(a), θy(a) up to scalar multiplication. Note that if the surface [θ] : M → P (V 4,∗) is defined in the affine coordinates by z = f(x, y), that is θ(x, y) = (1, x, y, f(x, y)), then the bilinear form β coincides with the Hessian of f at a. Again, there are several cases to distinguish: (a) β is indefinite, that is the signature of β is (1, 1); (b) β is definite; (c) β is degenerate. Let us consider case (a). At each point a ∈ M there exists a direct sum decomposition TaM = Ea⊕Fa wtih dimEa = dimFa = 1 such that β(v, v) = 0 for v ∈ Ea or v ∈ Fa (E and F are called asymptotic directions of θ). Then take local coordinates x, y so that x and y are 1st integrals of E and F respectively. With this choice of coordinates we get θxy 6≡ 0, θxx ≡ θyy ≡ 0 mod θ, θx, θy. 22 B. Doubrov, Y. Machida and T. Morimoto Therefore there exist functions ai, bi, ci of (x, y) (i = 1, 2) such that {θ0, θ1, θ2, θ3} is a funda- mental system of solutions of{ uxx = a1ux + b1uy + c1u, uyy = a2ux + b2uy + c2u. (Dii−a) Thus, we have shown that for a surface [θ] : M2 → P ( V 4,∗) of signature (1, 1) the differential equation corresponding to ϕ : M → Flag(V, φ) is equivalent to (Dii−a), where ϕ∗ is the osculating map generated by [θ] and ϕ is dual to ϕ∗. Conversely, suppose we are given a system of differential equations of the form (Dii−a). Note first that since all higher order derivatives uxxx, uxxy, . . . can be expressed via u, ux, uy, and uxy, this system is of finite type and the dimension of solution space is smaller or equal to 4. The equality holds only when all compatibility conditions are satisfied. If this is the case, then the surface in P ( R4,∗) defined by a fundamental system of solutions of (Dii−a) has signature (1, 1). Further study of the remaining cases (b) and (c) is left to the reader. See also [38] and [19, 29] for more detailed studies of case (a). The most symmetric case of the non-degenerate surface in P ( V 4,∗) is given by the Segre embedding P (U)× P (U∗) 3 ( [u], [α] ) 7→ [u⊗ α] ∈ P (U ⊗ U∗), where U is a 2-dimensional vector space. The determinant on U ⊗ U∗ = Hom(U,U) defines a quadratic form δ on U ⊗ U∗ of signa- ture (2, 2), and δ(u⊗ α) = 0, so that the surface is a quadric in P (U ⊗ U∗). Note that the group G = SL(U)× SL(U) acts on U ⊗ U∗ by ρ(a, b)v ⊗ α = av ⊗ b∗,−1α for v ∈ U, α ∈ U∗, a, b ∈ SL(U), and therefore acts on (U ⊗ U∗) = U∗ ⊗ U by ρ(a, b)∗,−1. Let {ε0, ε1} and { ε0, ε1 } be bases of U and U∗ dual to each other. Then the maps ω and θ defined by ρ(a, b) ( ε0 ⊗ ε1 ) ⊂ V = U ⊗ U∗ G = SL(U)× SL(U) 3 (a, b) ρ∗,−1(a, b) ( ε1 ⊗ ε0 ) ⊂ V ∗ ω θ give rise to the embeddings [ω]Seg : M → P (V ) and [θ]Seg : M → P (V ∗), where M = P (U)×P (U∗) and V = U⊗U∗. Then [ω]Seg and [θ]Seg generate osculating mappings respectively ϕSeg : M → Flag(V, φ) and ϕ∗Seg : M → Flag(V ∗, φ∗), which are dual to each other. We have θ (( ξ0 ξ1 ) ⊗ (η0, η1) ) = ( ε0, ε1 )((−ξ1 ) η1 ( −ξ1 ) (−η0) ξ0η1 ξ0(−η0) )( ε0 ε1 ) . Extrinsic Geometry and Linear Differential Equations 23 In affine coordinates x = ξ1/ξ0, y = η1/η0, the components of θ are {−xy, x, y,−1} and form a fundamental system of solutions of{ uxx = 0, uyy = 0. This is precisely the differential equation that corresponds to the Segre embedding ϕSeg : M2 → Flag ( V 4, φ ) . Example 4.9. Consider a submanifold [θ] : (M, f) → P (V ∗) represented by a smooth map θ : M → V ∗\{0}, where M is a 3-dimensional contact manifold endowed with the filtration f induced from a contact distribution D = f−1. To fix the notation we assume M = R3 with the standard coordinates (x, y, z) and the contact distribution defined by the contact form: ω = dz + 1 2 (x dy − y dx). Therefore the contact distribution is spanned by X = ∂ ∂x + 1 2 y ∂ ∂z , Y = ∂ ∂y − 1 2 x ∂ ∂z and { X,Y, Z ( = ∂ ∂z )} forms a moving frame on M . Note that [X,Y ] = −Z and we count w-ordX = w-ordY = 1, w-ordZ = 2 with respect to the weighted order associated with the contact filtration f. Suppose that θ : M → V ∗\{0} is regularly generating V ∗, that is there exists an osculating map ϕ∗ : (M, f)→ Flag(V ∗, φ∗) generated by θ and ϕ∗0 = 〈θ〉 ⊃ ϕ∗1 = 0. Recalling that 1, X, Y, Z,X2, XY, Y 2, XZ, Y Z,X3, X2Y, . . . form the basis of the ring D of differential operators on M , we may study possible type of φ∗ by ( dimφ∗0,dimφ∗−1, . . . ) . If we assume that θ, Xθ, Y θ are independent, then the first case to examine will be the case of type (1, 3, 4, . . . ). Suppose, for instance, that X2θ, XY θ, Y 2θ ≡ 0 mod (θ,Xθ, Y θ). Then the components of θ satisfy a system of differential equations of the following form: X2u = A1Xu+B1Y u+ C1u, Y 2u = A2Xu+B2Y u+ C2u, XY u = A3Xu+B3Y u+ C3u, (4.1) where Ai, Bi, Ci (i = 1, 2, 3) are functions on M . It is easy to see that {1, X, Y, Z} and D 〈 X2, XY, Y 2 〉 are complementary and generate D. Therefore the system (4.1) is of finite type with the dimension of the solution space ≤ 4. Therefore in this case we have dimV = 4 and the type of φ is (1, 3, 4). For example, if the right hand side of equation (4.1) vanishes identically, its solution space is spanned by functions {1, x, y, z + xy/2}. Next, consider the case of type (1, 3, 5, . . . ). If we assume X2θ, Y 2θ ≡ 0 mod (θ,Xθ, Y θ), we find an interesting class of differential equations{ X2u = A1Xu+B1Y u+ C1u, Y 2u = A2Xu+B2Y u+ C2u, (4.2) where Ai, Bi, Ci are functions on M . 24 B. Doubrov, Y. Machida and T. Morimoto It is easy to see that { 1, X, Y,XY,Z,XZ, Y Z,Z2 } are complementary to D 〈 X2, Y 2 〉 and generate the whole D together with it. Therefore the solution space of this system is ≤ 8. If the equality holds, then dimV = 8 and the type of φ∗ is (1, 3, 5, 7, 8). Now assume that θ : M → V ∗,8 satisfies the above conditions. Then in this case also by the same argument as in Example 4.7, we see that the differential equation corresponding to ϕ : M → Flag(V, φ) is equivalent to (4.2), where ϕ is the dual to ϕ∗ : M → Flag(V ∗, φ∗). The most symmetric model of ϕ : M → Flag ( V 8, φ ) is obtained by the adjoint representation of SL(3,R) on the Lie algebra sl(3,R) as shown in Section 6.5. 5 Equivalence problems, extrinsic normal Cartan connections and invariants In this section we study the equivalence problem of L/L0 extrinsic geometries and L/L0 diffe- rential equations via L/L0 extrinsic bundles. We will consider exclusively those objects that are of constant symbol, say of type (g−, grV,L). Otherwise, we have to deal with principal fibre bundles with varying structure groups as studied in [20]. The main idea here for the equivalence problem is, starting from a L/L0 extrinsic bundle, to construct, in a canonical manner, a nice reduction (a subbundle ) whose structure function represents effectively the invariants of the original structure. We shall single out the conditions for the symbol (g−, grV,L) under which we can construct what we call an extrinsic Cartan connection associated with the original structure. The general case in which it is no more possible to associate Cartan connection is treated in Section 7. 5.1 Relative prolongations, standard models and extrinsic cohomology groups As in Section 3, let (V, φ) be a filtered vector space, L ⊂ GL(V ) a Lie subgroup, L0 the Lie subgroup of L which fixes φ, l and l0 the Lie algebras of L and L0 respectively. Denote again by φ the induced filtration on gl(V ) and l. In this section we assume that: (C0) There exists a filtration preserving identification of V and grV that induces the isomor- phism of l ⊂ gl(V ) and gr(l, φ) ⊂ gl(grV ). Using these identifications, we will write V = ⊕ Vp, l = ⊕ lp, so that φpV = ⊕ i≥p Vi and φpl = ⊕ i≥p li. Let g− be a graded Lie subalgebra of l−. We define ḡ = ⊕ ḡp inductively as follows: For p < 0, ḡp = gp and for p ≥ 0, ḡp = { A ∈ lp : [A, g−] ⊂ ⊕ i<p ḡi } . Then as easily seen, ḡ is a graded Lie subalgebra of l. Definition 5.1. The graded Lie algebra ḡ defined above is called the relative prolongation of g− with respect to l and denoted Prol(g−, l). Define the following subgroups of L corresponding to subalgebras ḡ0, g−, ḡ0 respectively: Ḡ0 = { a ∈ L : Ad(a)g− ⊂ g−, and aVp ⊂ Vp, ∀p } , Ḡ− = exp(g−), Ḡ0 = Ḡ0 exp(⊕p>0ḡp). Extrinsic Geometry and Linear Differential Equations 25 For simplicity we assume that there exists a Lie subgroup Ḡ ⊂ L corresponding to the subalgebra ḡ ⊂ l and containing the subgroup Ḡ0 (and thus Ḡ0). Note that our main results do not depend on the existence of such subgroup and are based only on the subgroup Ḡ0 and the subalgebra ḡ. Let S be any subgroup of L such that G− = Ḡ− ⊂ S ⊂ Ḡ and S0 = S ∩ L0. Then the homogeneous space S/S0, endowed with the canonical invariant tangential filtration, is a standard filtered manifold of type g−. Definition 5.2. The induced map ϕmodelS : S/S0 → L/L0 ⊂ Flag(V, φ) is called a standard model immersion of type (g−, V, L). Note that all ϕmodelS are locally isomorphic and contain identical open orbit of the sub- group G−. So that we will not specify S and write ϕmodel unless needed. According to the categorical isomorphisms in Section 3, we may also speak of standard model of L/L0 differential equation, or extrinsic bundle, of type (g−, grV,L). In particular the principal fibre bundle Ḡ→ Ḡ/Ḡ0 equipped with the pull-back of the Maurer– Cartan form ωL is a standard model of L/L0 extrinsic bundle of type (g−, grV,L). Now we introduce the cohomology group which plays an important role in extrinsic geometry. The quotient space l/ḡ is a ḡ-module and a fortiori g−-module. There is the chain complex and the cohomology group Hp(g−, l/ḡ) associated with the representation of g− on l/ḡ: 0 −→ l/ḡ ∂−→ Hom (g−, l/ḡ) ∂−→ Hom ( ∧2g−, l/ḡ ) ∂−→ · · · . Noting that the coboundary operator preserves the filtration associated to φ, we actually consider the subcomplex 0 −→ φ1l/ḡ ∂−→ φ1 Hom (g−, l/ḡ) ∂−→ φ1 Hom ( ∧2g−, l/ḡ ) ∂−→ · · · . 5.2 Auxiliary groups, complementary subspaces For k ≥ 0 we define Lie subalgebras ḡ(k) of l by ḡ(k) = ḡ0 ⊕ · · · ⊕ ḡk ⊕ lk+1 ⊕ · · · and the Lie subgroups by Ḡ(k) = Ḡ0 exp ( φ1ḡ(k) ) , where φ1ḡ(k) = ḡ1 ⊕ · · · ⊕ ḡk ⊕ lk+1 ⊕ · · · = φ1l ∩ ḡ(k). By construction Ḡ(k) ⊂ Ḡ(k − 1) and Ḡ(k) = Ḡ0 for k bigger than the maximal i such that li 6= 0. Clearly the Lie algebra ḡ0 of Ḡ0 is φ0ḡ. We put E(k) = g− ⊕ ḡ(k), which are graded vector spaces converging to the Lie algebra ḡ. Taking an arbitrary complementary graded subspace ḡ′ = ⊕ ḡ′p of ḡ in l, we fix a direct sum decomposition l = E(k)⊕ E(k)′, where E(k)′ = ⊕ i≤k ḡ ′ i. 26 B. Doubrov, Y. Machida and T. Morimoto 5.3 First reduction, bundle Q(0) Let V , l, L, g− be as above. Let ϕ : (M, f)→ L/L0 ⊂ Flag(V, φ) be an osculating immersion of constant symbol of type (g−, grV,L). Let ( Q(−1), ω(−1) ) be the L/L0 extrinsic bundle corresponding to the L/L0 extrinsic geo- metry ϕ, or more precisely, Q(−1) is the induced bundle ϕ∗L of the bundle L → L/L0 by the map ϕ : M → L/L0 and ω(−1) = Φ(−1)∗ωL, where ωL denotes the Maurer–Cartan form of L and Φ(−1) : ϕ∗L→ L the canonical inclusion. Since ϕ is of constant symbol of type (g−, grV,L), so is Q(−1). Therefore for each x ∈ M there exists z ∈ Q(−1)x such that grωz(gr fx) ⊂ g−. Let Q(0) be the totality of such z. Then Q(0) is a principal fibre bundle overM with structure group Ḡ(0), and the pair (Q(0), ω(0)) proves to be a L/L0 extrinsic bundle congruent to Q(−1), where ω(0) is the pull-back of ω(−1). We may repeat the construction of (Q(0), ω(0)) more directly from the map ϕ as follows: For x ∈M we define Q(0)x to be the set of all filtration preserving isomorphism a : (V, φ)→ (V, ϕ(x)) such that a ∈ L and that gr a induces a module isomorphism (β, gr a) : (g−, grφ)→ ( gr fx, grϕ(x) ) . Note that β is uniquely determined by gr a as β(ξ) = (gr a)ρ(ξ)(gr a)−1 for ξ ∈ g−, where ρ denotes the representation of gr fx on grϕ(x). Put Q(0) = ⋃ x∈M Q(0)x, then we see that Q(0) is a principal fibre bundle over M with structure group Ḡ(0). Moreover the map Φ(0) : Q(0)→ L which sends a ∈ Q(0)x to a is an immersion lifting ϕ. Proposition 5.3. Let ω(0) (or simply ω) be the pull-back Φ(0)∗ωL. Then ω(0) is a 1-form on Q(0) taking values in l, and we have: (i) R∗aω = Ad ( a−1 ) ω, for a ∈ Ḡ(0). (ii) 〈Ã, ω〉 = A, for A ∈ ḡ(0). (iii) dω + 1 2 [ω, ω] = 0. (iv) Define the filtration of Q(0) as follows: for p ≤ 0, fpTQ being the inverse image of fpTM and for i > 0, fiTQ the induced filtration from φiE(0). Then for each z ∈ Q(0), the map ωz : TzQ(0)→ l is filtration preserving, and grωz : gr(TzQ(0), f)→ gr l maps gr(TzQ(0), f) isomorphically onto E(0) for any z ∈ Q. Proof. The assertions (i), (ii) and (iii) are clear. Let us prove (iv). Let π : Q(0)→M be the canonical projection, z ∈ Q(0), and x = π(z). Let g be a local cross- section of π : Q(0)→M around x with g(x) = z. Denote by ρ the representation of g− on grV and by θ that of gr fx on grϕ(x). Then by the definition of Q(0) there exists β : g− → gr fx such that it holds, for A ∈ gp, v ∈ grq φ, ρ(A)v = (gr g(x))−1θ(βA) gr g(x)v. If we take a local cross section X of ϕp representing ξ = β(A), we have (gr g(x))−1θ(βA) gr g(x)v ≡ g(x)−1Xx(gv) mod φp+q+1 = 〈ω, g∗Xx〉v ≡ 〈grω, ξ〉v mod φp+q+1. Extrinsic Geometry and Linear Differential Equations 27 Hence grω(ξ) = ρ ( β−1ξ ) for ξ ∈ gr fx, from which it follows that grω maps grTzQ(0) into E(0). But in view of the following commutative diagram and from our assumption that ϕ is an im- mersion, we see that grω : grTzQ(0)→ E(0) is an isomorphism: TxM g∗−−−−→ Tg(x)L ω−−−−→ L−1 g(x) TeL l∥∥∥ yπL∗ yπL∗ yπl TxM ϕ∗−−−−→ Tϕ(x)L/L 0 −−−−→ L̄−1 g(x) TēL/L 0 l/l0. � 5.4 Structure function χ According to the direct sum decomposition l = E(0)⊕ E(0)′, we write ω(0) = ω(0)I + ω(0)II , or simply ω = ωI + ωII , where ωI , ωII take values in E(0), E(0)′ respectively. Then by (iv) of the above proposition ωI : TzQ(0)→ E(0) is an isomorphism and ωII : TzQ(0)→ E(0)′ is a map of order 1 and vanishes on the vertical vectors. We define χ = χ(0) by ω(0)II = χ(0)ω(0)I , Then χ(0) is a function on Q(0) taking values in φ1 Hom (E(0), E(0)′). Since χ(0)A = 0 for A ∈ ḡ(0), it takes values in Hom (g−, E(0)′). Since we see Hom ( g−, E(0)′ ) ⊂ Hom ( g−, ḡ ′) ∼= Hom (g−, l/ḡ), identifying ḡ′ with l/ḡ, we may regard χ(0) as a function taking values in Hom(g−, l/ḡ) and actually in φ1 Hom(g−, l/ḡ). 5.5 Condition (C) Starting from Q(0), we carry a series of reductions by using the structure function χ. We make the following assumptions (C1), (C2) which are almost indispensable to construct an extrinsic Cartan connection. First, note that the adjoint action of Ḡ0 on l leaves invariant ḡ and φ0ḡ and acts naturally on l/ḡ and ḡ/φ0ḡ. We shall identify g− with ḡ/φ0ḡ and define the action of Ḡ0 on g− via this identification. Therefore Ḡ0 acts on Hom (g−, l/ḡ) and leaves invariant φ1 Hom (g−, l/ḡ). (C1) There exists a Ḡ0-invariant graded subspace ḡ′ ⊂ l such that l = ḡ⊕ ḡ′. (C2) There exists a Ḡ0-invariant graded subspace W = ⊕ Wp ⊂ φ1 Hom (g−, l/ḡ) such that φ1 Hom (g−, l/ḡ) = ∂φ1(l/ḡ)⊕W . Note that the representation ρ of Ḡ0 on Hom (g−, l/ḡ) induces the representation ρ(k) of Ḡ(k) = Ḡ0/φk+1Ḡ0 on Hom (g−, l/ḡ) /φk+1 Hom (g−, l/ḡ). If W is Ḡ0-invariant, then W (k) ( = W/φk+1W ) is Ḡ(k)-invariant. 28 B. Doubrov, Y. Machida and T. Morimoto 5.6 Reductions Now let us proceed to the construction of a series of L/L0 extrinsic bundles {Q(k), k ≥ 0}. Conducted by the following diagram : L LxΦ(k) xΦ(k+1) Hom(g−, l/ḡ) χ(k)←−−−− Q(k) ιk+1,k←−−−− Q(k + 1) χ(k+1)−−−−→ Hom(g−, l/ḡ)y y y y Hom(g−, l/ḡ)(k+1) χ(k)(k+1) ←−−−−−− Q(k)(k+1) ῑk+1,k←−−−− Q(k + 1)(k+1) χ(k+1)(k+1) −−−−−−−−→ W (k+1)y exp lk+1 y exp ḡk+1 y y W (k) χ(k)(k)←−−−− Q(k)(k) ∼=←−−−− Q(k + 1)(k) χ(k+1)(k)−−−−−−→ W (k)y yḠ0/φk+1 M M we are going to construct bundles with 1-forms ( Q(k), Ḡ(k),M ;ω(k) ) inductively for k = 0, 1, 2, . . . in such a way that the following conditions are satisfied. (k)1 Canonically associated to an immersion ϕ : (M, f) → L/L0 of type (g−, grV,L), there is a principal fibre bundle Q(k) over (M, f) with structure group Ḡ(k), and an immersion Φ(k) : Q(k)→ L which gives bundle homomorphism: Q(k) Φ(k)−−−−→ LyḠ(k) yL0 M ϕ−−−−→ L/L0. (k)2 There is an l-valued 1-form ω(k) defined by ω(k) = Φ(k)∗Ω satisfying R∗aω(k) = Ad ( a−1 ) ω(k) ∀a ∈ Ḡ(k),〈 ω(k), Ã 〉 = A ∀A ∈ ḡ(k), dω(k) + 1 2 [ω(k), ω(k)] = 0. Write ω(k) = ω(k)I + ω(k)II with ω(k)I and ω(k)II being E(k) and E(k)′-valued 1-forms respectively. Then the li- near map (ω(k)I)z : TzQ(k) → E(k) is a filtration preserving isomorphism and ω(k)II : TzQ(k)→ E(k)′ is a map of order 1 for each z ∈ Q(k). (k)3 Define a map χ(k) : Q(k)→ φ1 Hom(g−, l/ḡ) by ω(k)II = χ(k)ω(k)I , or symbolically χ(k) = ω(k)II ω(k)I . By definition χ(k) is a Hom(E(k), E(k)′)-valued function on Q(k), but by (k)2 it actually takes values in φ1 Hom(g−, E(k)′). Noting that E(k)′ ⊂ ḡ′ and identifying ḡ′ with l/ḡ, we regard χ(k) as a map from Q(k) to Hom(g−, l/ḡ). Extrinsic Geometry and Linear Differential Equations 29 (k)4 If k ≥ 1, then Q(k) = { z ∈ Q(k − 1) : χ(k − 1)(k)(z) ∈W (k) } . (k)5 χ(k)(k) = ι∗k,k−1χ(k − 1)(k), where ιk,k−1 denotes the canonical injection Q(k) → Q(k − 1). Therefore χ(k)(k) takes values in W (k) = W/φk+1W and R∗gχ(k)(k) = ρ ( g−1 ) χ(k)(k) for g ∈ Ḡ0. Note here that the action of Ḡ0 on φ1 Hom(g−, l/ḡ)/φk+1 only depends on Ḡ0/φkḠ0. (k)6 The map χ(k)(k+1) : Q(k)(k+1) → Hom(g−, l/ḡ)(k+1) satisfies R∗g expAk+1 χ(k)(k+1) = ρ ( g−1 ) χ(k)(k+1) + ∂ ◦ π(Ak+1) for g ∈ Ḡ0, Ak+1 ∈ φk+1g(k), where ∂ ◦ π is the composition of the following maps: l π−−−−→ l/ḡ ∂−−−−→ Hom(g−, l/ḡ). (k)7 Let χ(k)j denote the Hom(g−, l/ḡ)j-component of χ(k) and χ(k)(`) = ∑ j≤` χ(k)j . Then ∂χ(k)k+1 = Bk+1 ( χ(k), Dχ(k)(k) ) , where Bk+1 is a Hom ( ∧2g−, l/ḡ ) k+1 -valued function determined by χ(k)(k) and its deriva- tives Dχ(k)(k). In particular, if χ(k)(k) ≡ 0, then B(k) ≡ 0. Now let us prove the assertions above by induction on k. Supposing that for a non-negative integer k the assertions (j)1, . . . , (j)7 (j < k) hold, we prove the assertions for k. First for (k)1, if k = 0, Q(0) is already defined. If k ≥ 1, we define Q(k) by (k)4. Then we see from (k − 1)6 that for every x ∈ M there exists z in the fibre Q(k − 1)x over x such that χ(k − 1)(k)(z) ∈ W (k) and that for a ∈ Ḡ(k − 1) we have χ(k − 1)(k)(za) ∈ W (k), if and only if a ∈ Ḡ(k). Hence Q(k) is a principal fibre bundle over M with structure group Ḡ(k). Setting Φ(k) to be the restriction of Φ(k − 1) to Q(k), we finish the construction (k)1. To verify the assertion (k)2 we have only to note that ω(k)I = ι∗k,k−1 ( ω(k − 1)I − ω(k − 1)ḡ′k ) and ι∗k,k−1ω(k − 1)ḡ′k ≡ 0 mod ω(k)I , from which it follows that ω(k)I gives an isomorphism TzQ(k)→ E(k) at each z ∈ Q(k). Now χ(k) being defined by (k)3, let us prove (k)5. Let ι : Q(k)→ Q(k − 1) be the inclusion. We have ι∗ω(k − 1)I = ω(k)I + ι∗ω(k − 1)ḡ′k , ι∗ω(k − 1)II = ω(k)II − ι∗ω(k − 1)ḡ′k . Let v ∈ gp (p < 0), and take X ∈ TzQ(k) such that 〈ωI , X〉 = v. Then χ(k)(z)v = 〈ωII , X〉. But we have〈 ι∗ω(k − 1)I , X 〉 = v + 〈 ι∗ω(k − 1)ḡ′k , X 〉 . 30 B. Doubrov, Y. Machida and T. Morimoto If we put Bk = 〈ι∗ω(k−1)ḡ′k , X〉 ∈ lk and Y = X−B̃k ∈ TzQ(k−1), then we have 〈ω(k−1)I , Y 〉 = v and χ(k − 1)(z)v = 〈 ω(k − 1)II , Y 〉 = 〈 ω(k − 1)II , X − B̃k 〉 = 〈 ω(k − 1)II , X 〉 = 〈 ω(k)II − ι∗ω(k − 1)ḡ′k , X 〉 = χ(k)(z)v −Bk ∼= χ(k)(z)v mod φk+1+p, which implies: χ(k)(k) = χ(k − 1)(k). The last formula in (k)5 is a consequence of (k − 1)6. Now let us prove the assertion (k)6. Let a = g expA, where g ∈ Ḡ and A ∈ φk+1l of which lk+1 component is denoted Ak+1. By definition χ(k) = ω(k)II ω(k)I and we have R∗aχ(k) = R∗a(ω(k)II) R∗a(ω(k)I) = (R∗aω(k))II (R∗aω(k))I = ( Ad ( a−1 ) ω(k) ) II( Ad ( a−1 ) ω(k) ) I . Now we have( Ad ( a−1ω )) I = ( Ad(expA)−1Ad ( g−1 ) ω ) I ≡ ( Ad ( g−1 ) ω − [ A,Ad ( g−1 ) ω ]) I mod Order(2k + 2) = ( Ad ( g−1 ) (ωI + ωII)− [ A,Ad ( g−1 ) (ωI + ωII) ]) I ≡ Ad ( g−1 ) ωI − ([ A,Ad ( g−1 ) (ωI) ]) I mod Order(k + 2) ≡ Ad ( g−1 ) ωI mod Order(k + 1). We have also( Ad ( a−1ω )) II = ( Ad(expA)−1Ad ( g−1 ) ω ) II ≡ ( Ad ( g−1 ) ω − [ A,Ad ( g−1 ) ω ]) II mod Order(2k + 2) = ( Ad ( g−1 ) (ωI + ωII)− [ A,Ad ( g−1 ) (ωI + ωII) ]) II ≡ Ad ( g−1 ) ωII − [ A,Ad ( g−1 ) (ωI) ] II mod Order(k + 2). Now let p < 0 and v ∈ gp. Take X,Y ∈ TzQ(k) satisfying〈 Ad ( g−1 ) (ωI), X 〉 = 〈( Ad ( a−1 ) ω ) I , Y 〉 = v. Put Z = Y − X. Then from the above formula it follows that the map X 7→ Z is of order ≥ k + 1. Then we have: (R∗aχ) v = 〈( Ad ( a−1 ) ω ) II , Y 〉 = 〈( Ad ( a−1 ) ω ) II , X + Z 〉 ≡ 〈( Ad ( g−1 ) ω ) II − [ A,Ad ( g−1 ) ωI ] II , X + Z 〉 ≡ Ad ( g−1 ) χ(Ad(g)v)− [A, v]II mod Order(k + 2). Hence we have R∗aχ (k+1)v = ρ ( g−1 ) χ(k+1)v − [Ak+1, v]II , which proves (k)6. Finally let us verify (k)7. From the structure equation we have dωI + 1 2 [ω, ω]I = 0, dωII + 1 2 [ω, ω]II = 0, ωII = χωI . Extrinsic Geometry and Linear Differential Equations 31 Hence we have dωI + 1 2 [ωI , ωI ]I + [ωI , ωII ]I + 1 2 [ωII , ωII ]I = 0, (5.1) dωII + [ωI , ωII ]II + 1 2 [ωII , ωII ]II = 0. (5.2) Let v ∈ gp, w ∈ gq (p, q < 0) and X, Y be the vector fields on Q(k) determined by 〈ωI , X〉 = v, 〈ωI , Y 〉 = w. Evaluating the equations on X, Y , we have −〈ωI , [X,Y ]〉+ [v, w] +A ( [v, χ(w)]I ) + [χ(v), χ(w)]I = 0, (5.3) Xχ(w)− Y χ(v)− 〈ωII , [X,Y ]〉+ [v, w]II +A ( [v, χ(w)]II ) + [χ(v), χ(w)]II = 0, (5.4) where A denotes the alternating sum in v, w. Substituting (5.3) into (5.4), we get Xχ(w)− Y χ(v)− χ ( [v, w] +A ( [v, χ(w)]I ) + [χ(v), χ(w)]I ) + [v, w]II +A ( [v, χ(w)]II ) + [χ(v), χ(w)]II = 0. (5.5) Let us compute (5.5) under mod φp+q+k+2. Observing that Xχk+1(w) ∈ φk+1+q ⊂ φp+q+k+2, we have −χk+1([v, w]) +A ( [v, χk+1(w)]II ) = −Xχ(k)(w) + Y χ(k)(v) + χ(k) ( [v, w] +A ([ v, χ(k)(w) ] I ) + [ χ(k)(v), χ(k)(w) ] I ) −A ([ v, χ(k)(w) ] II ) − [ χ(k)(v), χ(k)(w) ] II . (5.6) Thus we have ∂χk+1 = Bk+1 ( χ(k), Dχ(k) ) , (5.7) where Bk+1 is a Hom ( Λ2g−, l/ḡ ) -valued function defined by the right hand side of (5.6). Thus we have proved the assertions (k)1, . . . , (k)7, completing the inductive construction of ( Q(k), ω(k), χ(k) ) . Let (P, ω, χ) to be ( Q(k), ω(k), χ(k) ) for k bigger than the maximal i such that li 6= 0. Let us now formulate the main result of this section. Definition 5.4. We say that an L/L0 extrinsic bundle π : P → (M, f) is an extrinsic W -normal Cartan connection, if its structure group is Ḡ0 ⊂ L0 and the structure function χ defined by ωII = χωI takes values in the subspace W . Here we write ω = ωI + ωII according to the direct sum decomposition l = ḡ + ḡ′. Note that ωI is an absolute parallelism on P , which defines a Cartan connection on M modelled by Ḡ/Ḡ0 in the usual sense. We also note that the notion of extrinsic W -normalily does not depend on the choice of the space ḡ′. Theorem 5.5. Let {V, φ, L, g−} be as above and satisfy (C0), (C1) and (C2). Then for each immersion ϕ : (M, f)→ L/L0 ⊂ Flag(V, φ) of type (g−, V, L) we can construct canonically an ext- rinsic W -normal Cartan connection (P, ω). In particular, two immersions (M,ϕ) and (M ′, ϕ′) are L-equivalent if and only if the corresponding extrinsic Cartan connections (P, ω) and (P ′, ω′) are isomorphic. 32 B. Doubrov, Y. Machida and T. Morimoto Corollary. The structure function χ along with its derivatives fulfills completely the invariants of (P, ω). In particular, the immersion (M, f)→ L/L0 ⊂ Flag(V, φ) of type (g−, V, L) is locally equivalent (under the action of L) to the model embedding if and only if χ = 0. Remark 5.6. Of course the structure function χ : P → Hom (ḡ, l/ḡ) depends on the choice of a complementary subspace ḡ′ to ḡ in l. Let ḡ′′ be another Ḡ0-invariant complementary subspace and let χ′ be the structure function determined by this choice. Then we see easily that χ′ = (1− χλ)−1χ = ( 1 + χλ+ 1 2! (χλ)2 + · · · ) χ, where λ ∈ Hom(l/ḡ, ḡ) is given by λ = σ′−σ, σ, σ′ being the isomorphisms from l/ḡ to ḡ′ and ḡ′′ respectively. In particular, if χ vanishes identically up to order k, then the same holds also for χ′. Similarly, the structure function χ also depends on the choice of the complementary sub- space W : φ1 Hom (ḡ, l/ḡ) = φ1∂(l/ḡ)⊕W. Let W ′ be another Ḡ0-invariant complementary subspace and let χ′ be the new structure func- tion. Again, it can be easily shown that χ vanishes up to order k if and only if the same holds for χ′. See Section 7 for a generalization of the theorem and the corollary above. 6 Rigidity of rational homogeneous varieties 6.1 Extrinsic parabolic geometries Let g = ⊕µ i=−µ gi be a semi-simple graded Lie algebra. Denote by g− the nilpotent subalgebra⊕ i<0 gi and by p the parabolic subalgebra ⊕ i≥0 gi in g. Let V be an arbitrary faithful finite- dimensional irreducible representation of g. Let B be the Killing form of g. The following properties are well-known [2]: 1. There exists a grading element E ∈ g, such that gk = {X ∈ g \| [E,X] = kX}. 2. B(gi, gj) = 0 for all i+ j 6= 0, and the restriction of B to gi × g−i is non-degenerate. 3. In the real case, there exists an involution θ of g such that θ(gi) = g−i and the bilinear form (X,Y ) = −B(X, θ(Y )) is positive definite. Moreover, for any g-module V there exists a scalar product ( , ) on V such that (Xu, v) + (u, θ(X)v) = 0 for all u, v ∈ V, X ∈ g. 4. In the complex case, there exists an anti-involution σ of g such that σ(gi) = g−i and the Hermitian form (x, y) = −B(x, σ(y)) is positive definite. Moreover, for any g-module V there exists a positive definite Hermitian form ( , ) on V such that (Xu, v) + (u, σ(X)v) = 0 for all u, v ∈ V, X ∈ g. Extrinsic Geometry and Linear Differential Equations 33 Indeed, consider first the complex case. Then the existence of an anti-involution σ such that σ(gi) = g−i and the Hermitian form (x, y) = −B(x, σ(y)) is positive definite, is shown in [36, Lemma 1.5]. In particular, the corresponding real form gσ is compact. Hence, it stabilizes a certain positive-definite Hermitian form ( , ) on V . Since any element X ∈ g can be represented as X = X1 + iX2 and σ(X) = X1 − iX2, where X1, X2 ∈ gσ, we immediately get (Xu, v) + (u, σ(X)v) = (X1u, v) + (u,X1v) + (iX2u, v) + (u,−iX2v) = (X1u, v) + (u,X1v) + i(X2u, v) + i(u,X2v) = 0. Similarly, in the real case, the existence of an involution θ such that θ(gi) = g−i and the bilinear form (x, y) = −B(x, θ(y)) is positive definite, is also demonstrated in [36]. Let g = gθ(1)⊕ gθ(−1) be the eigenvalue decomposition of θ. Then it is well known that the Lie algebra gθ(1) + igθ(−1) is compact in gl(V C). So, it preserves a positive definite Hermitian form on V C. Restricting the real part of this form on V gives us the requires positive definite scalar pro- duct on V . For any A ∈ gl(V ), denote by A∗ the operator adjoint to A with respect to scalar (Hermitian) product ( , ) on V . Then the above conditions imply that θ(X) = −X∗ (resp., σ(X) = −X∗) for any X ∈ g. In particular, g is stable with respect to the (anti-)involution A 7→ −A∗ of gl(V ). Decomposing V according to the eigenvalues of the grading element E, we can equip V with the grading V = ⊕ Vj such that gi.Vj ⊂ Vi+j for all i, j. Note that the degrees in the sum V = ⊕ Vj may be rational, but the difference between any two degrees j, j′ for which both Vj and Vj′ are non-trivial is necessarily an integer. Let jmax be the largest such degree. Shifting the grading of V by −jmax−1, we can always normalize the grading of V such that it is concentrated in the negative degree with V−1 6= 0. Note that independently of such shift of degrees the induced grading of gl(V ) is always defined via eigenvalues of the grading element E. Similarly, if V is an arbitrary finite-dimensional representation of the Lie algebra g, we can decompose it into the sum of irreducible representations, introduce the above grading on each of the summands, and combine them into the grading of V . So, from now on we assume that V is an arbitrary (not necessarily irreducible) finite-dimensional representation of g, equipped with the above grading. Let φ be the induced filtration of V , which is by definition stabilized by p. Let G be a connected semisimple Lie group with a Lie algebra g, and let P be the parabolic subgroup corresponding to the subalgebra p. The homogeneous flag variety G/P is a ratio- nal homogeneous variety. Assume that the representation g → gl(V ) can be extended to the representation G→ GL(V ) of the Lie group G. Let ϕmodel : G/P → Flag(V, φ) be the standard embedding of type (g−, V,G). Lemma 6.1. Suppose that V is isotypic, that is all highest weights of V coincide. Then the relative prolongation Prol(g−, V ) is equal to g+Z(g) ⊂ gl(V ), where Z(g) is the centralizer of g in gl(V ). Proof. It is sufficient to prove lemma in the complex case, as the real case immediately follows via the complexification argument. Consider g-module gl(V ). Assume first that V is irreducible. As Prol(g−, V ) contains g, it is a submodule of the g-module gl(V ). Let U be any graded submodule of Prol(g−, V ) such that g ∩ U = 0. By definition U is concentrated in the non- negative degree. Hence, it generates a g-invariant subalgebra n that is also concentrated in a non-negative degree. In particular, g+n is also a subalgebra. Since g-module V is irreducible, (g + n) also acts irreducibly on V . Hence (see [1, Chapitre 1, Section 6, Théorème 4]), g + n is a reductive subalgebra in gl(V ). From the general theory of graded semisimple Lie algebras it follows that the dimensions of subspaces of opposite degrees should coincide. This is only 34 B. Doubrov, Y. Machida and T. Morimoto possible if n, and thus U is concentrated in degree 0. As [g−, U ] ⊂ U , this immediately implies that [g−, U ] = 0, and U ⊂ Z(g). Let V be now an arbitrary isotypic g-module, that is V is isomorphic to V ′⊗Ck, where V ′ is irreducible and Ck is a trivial g-module. We note that g-module gl(V ) is isomorphic to gl(V ′)⊗ gl(k,C) and any its irreducible submodule has the form U ⊗ L, where U is an irreducible submodule of gl(V ′) and L is a 1-dimensional subspace in gl(k,C). If such submodule lies in Prol(g−, V ) and has zero intersection with g, then it is concentrated in the non-negative degree. It is easy to see that U itself lies in Prol(g−, V ′). By above this means that U ⊂ Z(g). This completes the proof. � Remark 6.2. The condition of V to be isotypic can not be dropped. The simplest counter- example is given by g = sl(2,C) and V = V1 ⊕ V2, where V1 and V2 are two irreducible repre- sentations of sl(2,C) of different dimension. In the following we always assume that the g-module V is isotypic. We also say that ϕmodel is a standard embedding of G/P corresponding to the g-module V . 6.2 Harmonic theory on semisimple Lie algebras Let us introduce an invariant symmetric bilinear form on gl(V ) as follows Tr(A,B) = trAB, for all A,B ∈ gl(V ). It is clear that the restriction of Tr to gli(V ) × glj(V ) is identically zero for i + j 6= 0 and is non-degenerate for i+ j = 0. Note that the restriction of Tr to any semisimple subalgebra of gl(V ) is non-degenerate. In particular, the restriction of Tr to g is non-degenerate, and we have gl(V ) = g+g⊥, where g⊥ is an orthogonal complement of g in gl(V ) with respect to Tr. Similarly, let ḡ be the prolongation of (g−, V ). Then according to Lemma 6.1, we have ḡ = g + Z(g). Lemma 6.3. We have the following orthogonal decomposition with respect to Tr: gl(V ) = g⊕ Z(g)⊕ ḡ⊥. Proof. It is sufficient to consider only the complex case. As above, decompose V as V ′ ⊗ Ck, where the module V ′ is irreducible. Then from Schur’s lemma, we know that Z(g) = gl(k,C). Thus, it is easy to see that the restriction of Tr on Z(g) is non-degenerate. Moreover, for any X ∈ g and A ∈ Z(g), we have tr(XA) = tr(X) tr(A) = 0. � Note that all three components of this decomposition are g- and Z(g)-invariant. Moreover, since the Lie algebra g is graded and Z(g) ⊂ gl0(V ), we see that this decomposition (namely, the subspace ḡ⊥) is also compatible with the grading of gl(V ). Let ( C = ∑ q C q, ∂ ) be the cochain complex associated with the g−-module gl(V ). It inherits the grading from the gradings of g− and gl(V ): Cq = ∑ p Cqp , Cqp = { c ∈ Cq : c(g−i1 ∧ · · · ∧ g−iq) ⊂ glp−i1−···−iq(V ) } . It is clear that the coboundary operator ∂ preserves grading and, hence, maps Cqp to Cq+1 p . Similarly, for any g−-invariant graded subspace W in gl(V ), denote by C(W ) = ∑ Cqp(W ) the cochain complex associated with the g−-module W . We shall be particularly interested in C ( ḡ⊥ ) . Any cochain c ∈ Cq can be uniquely decomposed into the sum c = cI + cII according to the decomposition gl(V ) = ḡ⊕ ḡ⊥. Extrinsic Geometry and Linear Differential Equations 35 The cohomology Hq p ( g−, ḡ ⊥) can be effectively computed using Kostant theorem [16] and the notion of normality formalized by Tanaka [36]. Namely, we can extend the scalar products on g and V introduced in Section 6.1 to the spaces Cq and introduce the adjoint coboundary operator ∂∗ : Cq+1 → Cq and the Laplacian ∆ = ∂∂∗+∂∗∂ : Cq → Cq. The standard argument shows that ker ∆ = ker ∂ ∩ ker ∂∗ and we have the direct product decomposition Cq = im ∂⊕ im ∂∗⊕ ker ∆. We shall call an element c ∈ Cq harmonic, if ∆c = 0, which is equivalent to ∂c = ∂∗c = 0. Additionally, for any χ ∈ Cq we denote by Hχ the projection of χ to ker ∆ in the above direct product decomposition. We shall also call Hχ the harmonic part of χ. 6.3 Normal reductions Let now (M, f) be an arbitrary filtered manifold of constant type m = g−. Let ϕ : (M, f) → Flag(V, φ) be arbitrary osculating map. Define ḡ′ = ḡ⊥ in the condition (C1) and W = φ1 ker ∂∗ in the condition (C2) of Section 5.6. In particular, an extrinsic Cartan connection associated with the embedding ϕ is W -normal, if and only if we have ∂∗χ = 0. We shall call such extrinsic Cartan connections just normal. Applying Theorem 5.5, we immediately get Theorem 6.4. For any embedding ϕ : (M, f) → Flag(V, φ) of type (g−, V ), there is a unique normal extrinsic Cartan connection ω : TP → gl(V ) on the principal G 0 -bundle P →M . 6.4 Vanishing of cohomology groups and rigidity According to corollary of Theorem 5.5, the structure function χ = ∑ k≥1 χk : P → Hom ( g−, ḡ ⊥) k is a complete invariant of the embedding ϕ. Using Section 6.2, we identify H1 ( g−, ḡ ⊥) with ker ∆ = ker ∂ ∩ ker ∂∗ and consider Hχ as an element of H1 ( g−, ḡ ⊥). It is the fundamental invariant of the embedding ϕ in the following sense. Proposition 6.5. We have χ = 0 if and only if Hχ = 0. And in this case the embedding ϕ is locally equivalent to the standard embedding of type (g−, V ): ϕmodel : G/G0 → L/L0 ⊂ Flag(V, φ). Proof. Indeed, assume that for some k ≥ 0 we have χi = 0 for all i ≤ k. Then equation (5.7) from the proof of Theorem 5.5 implies also that ∂χk+1 = 0. Since χ is normal, we also have ∂∗χk+1 = 0. So, χk+1 = Hχk+1. By induction, we get that χ = 0 if and only if Hχ = 0. � In particular, if the cohomology spaces H1 r ( g−, ḡ ⊥) vanish identically for all r ≥ 1, then any embedding ϕ of type (g−, V ) is locally equivalent to the standard one. We say in this case that the standard embedding ϕmodel is rigid. So, we arrive at a natural question, for which irreducible representations the positive part of the cohomology H1 + ( g−, ḡ ⊥) does not vanish. By the following proposition, the only rational homogeneous varieties of simple Lie groups, which might be non-rigid are limited to (i) projective spaces, (ii) quadrics, (iii) A-type (Lagrange) contact spaces, (iv) C-type (projective) contact spaces. Proposition 6.6. Let g = ⊕ gp be a simple graded Lie algebra and U an irreducible submodule of the g-module gl(V ). The cohomology group H1 r (g−, U) vanishes for r ≥ 1 except for the following cases: 1. (A3,Σ) with Σ = {α2}, (Al,Σ) (l ≥ 1) with Σ = {α1}, {αl}, or {α1, αl}, 36 B. Doubrov, Y. Machida and T. Morimoto 2. (B2,Σ) with Σ = {α1}, or {α2}, (Bl,Σ) (l ≥ 3) with Σ = {α1}, 3. (Cl,Σ) (l ≥ 3) with Σ = {α1}, 4. (D4,Σ) with Σ = {α1}, {α3}, or {α4}, (Dl,Σ) (l ≥ 5) with Σ = {α1}, where Σ denotes the subset of the simple roots {α1, . . . , αl} which defines the gradation of g, that is, gα ⊂ gp for α = a1α1 + · · ·+ alαl if ∑ αi∈Σ ai = p. Proof. Let µ be the lowest weight of U . It is written as a linear combination of the fundamental weights: µ = ∑l i=1 µiωi with non-positive integers µi, where ωi is the i-th fundamental weight. By the theorem of Kostant [16], we have H1(g−, U) = ⊕ α∈Σ Lg0(−σα.(−µ)), where Lg0(−σα.(−µ)) denotes a g0-irreducible module of lowest weight: −σα.(−µ) = −(σα(−µ+ ρ)− ρ) = σα(µ− ρ) + ρ, σα being the reflection with respect to α and ρ = ω1 + · · · + ωl. Since σαi(ωj) = ωj − δijαi, we have σαi(µ− ρ) + ρ = σαi(µ) + αi. Now let us compute the degree of this weight. Since the grading of U is determined by the eigenvalues of the grading element E ∈ g, the degree of λ = ∑ λjαj is defined as deg(λ) = ∑ αi∈Σ λi. For αi ∈ Σ we have deg(σαi(µ) + αi) = deg(σαi(µ)) + 1. Let C−1 = (pij) be the inverse of the Cartan matrix C. Note that its components pij are all positive. We have σαi(µ) = σαi ( l∑ j=1 µjωj ) = l∑ j=1 µjωj − µiαi = l∑ j,k=1 µjpjkαk − µiαi. Therefore we have deg(σαi(µ)) = l∑ j=1 µj ∑ k′ pjk′ − µi = ∑ j 6=i µj ∑ k′ pjk′ + µi (∑ k′ pik′ − 1 ) , where the summation ∑ k′ is taken over all k′ such that αk′ ∈ Σ. We then deduce from this that if (g,Σ) has the following property:∑ αk∈Σ pik − 1 > 0 for all i such that αi ∈ Σ, then H1 r (g−, V ) = 0 for r ≥ 1. Now the theorem follows immediately from the table of (pij) as found, for instance, in Humphreys’ book [11, p. 69]. � Extrinsic Geometry and Linear Differential Equations 37 Consider now the cases from Proposition 6.6 in more detail. Let p ⊂ g be the parabolic subal- gebra in the complex simple Lie algebra that corresponds to one of the cases in Proposition 6.6. Considering p up to Aut(g), is equivalent to treating Σ up to the automorphism of the Dynkin diagram of the corresponding root system. Taking this into account, we may reduce the number of cases to one of the following: 1. (Al, {α1}), l ≥ 1, 2. (Bl, {α1}), l ≥ 2, 3. (Cl, {α1}), l ≥ 2, 4. (Dl, {α1}), l ≥ 4, 5. (A3, {α2}), 6. (Al, {α1, αl}), l ≥ 2. We use here the identification of (B2, {α2}) with (C2, {α1}). In the case (Al, {α1}), we get deg(σα1(µ)) = l∑ j=2 pj1µj + µ1(p11 − 1) = 1 l + 1 ( l∑ j=2 (l + 1− j)µj − µ1 ) ≥ 0. This gives an explicit condition on the lowest weight µ of the module U , such that H1(g−, U) is concentrated in the positive degree. In Bl, Cl, and Dl cases, we note that p11 = 1. So, we get deg(σα1(µ)) = l∑ j=2 pj1µj + µ1(p11 − 1) = l∑ j=2 pj1µj ≥ 0. Since µj ≤ 0, we immediately get that necessarily µ2 = · · · = µl = 0, and under this condition, H1(g−1, U) is concentrated in degree 1. Similarly, in the case (A3, {α2}), we have p22 = 1. So, we get that H1(g−1, U) is concentrated in positive degree if any only if µ1 = µ3 = 0, and this degree is equal to 1 in this case. Finally, in the case (Al, {α1, αl}), we get deg(σα1(µ)) = l∑ j=2 (pj1 + pjl)µj + µ1(p11 + p1l − 1) = l∑ j=2 µj ≥ 0. So, again we get µ2 = · · · = µl = 0 as the necessary condition for the module Lg0(−σα1 .(−µ)) ⊂ H1(g−, U) to be concentrated in the positive degree, which is equal to 1 in this case. Similarly, Lg0(−σαl .(−µ)) ⊂ H1(g−, U) is concentrated in positive degree (equal to 1) only for modules with the lowest weight µlωl, µl ≤ 0. Summing up, we get Proposition 6.7. Let µ = µ1ω1 + · · · + µlωl, µi ≤ 0, be the lowest weight of an irreducible module U over one of the simple Lie algebras of rank l from Proposition 6.6. Suppose H1(g−, U) has non-trivial components in positive degree. Then 1. In the case (Al, {α1}), we have N = l∑ j=2 (l + 1− j)µj − µ1 ≥ 0, where N is a multiple of l + 1. Then H1(g−, U) is an irreducible g0-module concentrated in degree N/(l + 1) + 1. 38 B. Doubrov, Y. Machida and T. Morimoto 2. In the cases Bl (l ≥ 2), Cl (l ≥ 2), Dl (l ≥ 4) and Σ = {α1}, and for A3, Σ = {α2}, we have µi = 0 unless αi ∈ Σ. Then H1(g−, U) is an irreducible g0-module concentrated in degree 1. 3. In the case (Al, {α1, αl}), we have µ2 = · · · = µl = 0 or µ1 = · · · = µl−1 = 0. Each of these conditions leads to an irreducible g0-module in H1(g−, U) concentrated in degree 1. All other cases from Proposition 6.6 are equivalent to the above via the automorphisms of the Dynkin diagram of the corresponding root system. 6.5 Simple Lie algebras of rank 2 and their adjoint representations In view of Propositions 6.6 and 6.7 let us consider, as first examples of extrinsic parabolic geometry on non-trivial filtered manifolds, the following simple Lie algebras of rank 2 with contact type gradings and their adjoint representations: (i) (A2, {α1, α2}), µ = ω1 + ω2, (ii) (C2, {α1}), µ = 2ω1, (iii) (G2, {α2}), µ = ω2, where {αi} indicates the grading, and µ is the highest weight of the adjoint representation. Throughout this subsection g (= ⊕gp) will denote one of the above simple Lie algebras with a contact type grading, that is, the negative part g− of g is a Heisenberg Lie algebra, or more precisely, g− = g−2 ⊕ g−1, dim g−2 = 1, and the bracket g−1 × g−1 → g−2 is non-degenerate. In the cases (i), (ii) dim g− = 3 and 5 in the case (iii). The representation ρ : g→ gl(V ) that we consider here is the adjoint representation, so that V = g, while we give a grading ⊕Vi on V by shifting: Vi = gi−2, to adjust to the usual orders of related differential equations. We fix a filtration φ of V by setting φi = ⊕p≥iVp. In each case we are going to inspect an osculating map ϕ : (M, f)→ L/L0 ⊂ Flag(V, φ) and the associated differential equation of symbol type (g−, V, L) with a suitable choice of a Lie subgroup L of GL(V ). Note that the filtered manifold (M, f) has g− as symbol, and therefore is a contact manifold of dimension 3 or 5. Consider in detail case (ii). We have g = sp(U), where U is a 4-dimensional vector space equipped with a non-degenerate symplectic form. We endow g with a contact type grading so that dim g±2 = 1, dim g±1 = 2, dim g0 = 4. As already said, V = g with Vi = gi−2 as a graded g module. We assume also that G = Sp(U). Consider an osculating map ϕ : (M, f)→ L/L0 ⊂ Flag(V, φ) for a Lie subgroup L of GL(V ) containing G. Then it is generated by the projective embedding ϕ4 : (M, f)→ P (V ). In particular the standard model embedding ϕmodel : G/G0 → Flag(V, φ) Extrinsic Geometry and Linear Differential Equations 39 is generated by the projective embedding ϕ4 model : G/G0 → P (V ), which may be called the contact Veronese embedding: It is isomorphic to the following projective embedding P (U)→ P ( S2(U) ) . Recall here that sp(U) may be identified with S2(U). As a choice of subgroup L ⊂ G we can take O(V ) (= O(V, κ)), where κ is the Killing form of g. So, we have natural embeddings G ⊂ O(V ) ⊂ GL(V ), g ⊂ o(V, κ) ⊂ gl(V ). Note that in the real case we have o(V, κ) = o(6, 4). It is easy to check that the relative prolongations of g− in o(V ) and gl(V ) coincide with g and ḡ = g + z respectively. Let us compute cohomology groups H1 +(g−, o(V )/g) and H1 + (g−, gl(V )/ḡ). In terms of g- modules, we have the irreducible decomposition gl(V ) = gl(10) = Γ0,0 ⊕ Γ0,1 ⊕ Γ2,0 ⊕ Γ0,2 ⊕ Γ2,1 ⊕ Γ4,0. Here l = o(6, 4) = Γ2,0 ⊕ Γ2,1. For the relative prolongation we have Prol(g−, o(6, 4)) = g, so ḡ = g = Γ2,0 = sp(4), ḡ⊥ = Γ2,1. Then by Proposition 6.7, we have no fundamental invariants of ϕ: H1 + ( g−, g ⊥) = 0. On the other hand, if we take l to be the whole gl(10), the relative prolongation of g− becomes Prol(g−, gl(10)) = g⊕ z = ḡ. Then H1 + ( g−, ḡ ⊥) = H1 1 (g−,Γ4,0) = S5R2 (dim = 6). It corresponds to the highest weight −α1 + 2α2 (and the lowest weight α1 − 2α2). The model equation can be written in this case as X3u = 0, XY Xu = 0, Y XY u = 0, Y 3u = 0 with X = ∂ ∂x + 1 2 y ∂ ∂z , Y = ∂ ∂y − 1 2 x ∂ ∂z for a contact coordinate system (x, y, z) of G/G0: The contact distribution D is defined by a contact form θ = dz − 1 2 (ydx− xdy), therefore X, Y are sections of D. 40 B. Doubrov, Y. Machida and T. Morimoto Now consider its deformed version X3u = aY 2u, XY Xu = 0, Y XY u = 0, Y 3u = 0, (Ea) where a is an arbitrary constant. We can verify that it still has a 10-dimensional solution space spanned by 1, x, y, z, x2, xy, xz, y2 + a 3 x3, yz − a 24 x4, z2 + a 120 x5, which defines an explicit (local) deformation of the standard embedding. Let us prove that this deformation is non-trivial. Proposition 6.8. If a 6= 0, then (Ea) is not locally equivalent to (E0) as linear differential equations. Proof. Let us show that the structure function χ = χ1 of an extrinsic bundle Q associated to the equation (Ea) does not vanish if a 6= 0, which will then prove by Theorem 5.5 that (Ea) is not isomorphic to (E0) if a 6= 0. We first note that since the principal parts of (Ea) are the same for all a, the symbol of (Ea) is isomorphic to that of the model equation and is equal to (g−, V ) as graded module, where g = sp(4,R) and V = g as graded module (modulo the shift of the grading). Next, let us note that (Ea) satisfies all compatibility conditions for any a, as suggested by the computer computation mentioned above. Let X, Y be as above and let Z = −[X,Y ] = ∂ ∂z . By simple computation, we have{ X2Y = −XZ, Y 2X = −Y Z, and for the 4-th order X4 = −aY Z, X3Y = 0, X2Y 2 = 1 2 Z2, XY 3 = 0, Y 4 = 0, X2Z = 0, XY Z = −1 2 Z2, Y 2Z = 0. The value of Z2 remains free. Extrinsic Geometry and Linear Differential Equations 41 For the 5-th order we have X5 = a 2 Z2, and all other 5-th order derivatives X4Y, X3Y 2, . . . X3Z, X2Y Z, . . . XZ2, Y Z2 vanish. We then see easily that all derivatives of order higher than 5 vanish. Thus, the equation (Ea) is prolonged to a weighted involutive system of order 5 and has a 10-dimensional solution space. Now note that since the symbol of (Ea) is isomorphic to (g−, grV ), this means that (grV )∗ ∼= U(g−)⊗ (gr−2 V )∗/I, where U(g−) denotes the universal enveloping algebra of g− and I denotes the left U(g−)-module generated by the generators of model equation (E0), that is X3 ⊗ Z∗, Y 3 ⊗ Z∗, XY X ⊗ Z∗, Y XY ⊗ Z∗, where Z∗ denotes the basis of (gr−2 V )∗ dual to Z. Recalling our convention of grading, e.g., grq V ∗ = (gr−q V )∗, we note that the isomorphism above is an isomorphism of g−-modules, which can be given explicitly as follows. Let {A1, . . . , A10} be a basis of sp(4,R) and {A∗1, . . . , A∗10} its dual basis of sp(4,R)∗ given by A1 = E14, A4 = E23, A7 = E32, A10 = E41. A2 = 1√ 2 (E13 + E24), A5 = 1√ 2 (E11 − E44), A8 = 1√ 2 (E21 − E43), A3 = 1√ 2 (E12 − E34), A6 = 1√ 2 (E22 − E33), A9 = 1√ 2 (E31 + E42), We identify X = A8, Y = A9, Z = A10. Then we can easily verify that the following map gives a g−-module isomorphism  A∗1 A∗2 ... A∗10  7→  −1 2Z 2 Y Z XZ −Y 2 − 1√ 2 Z − 1√ 2 (XY + Y X) X2 Y X 1  · Z∗. We denote by P the 10 by 1 matrix of differential operators that appear in the right hand side of the above correspondence. Let {θ1, . . . , θ10} be a fundamental system of solutions of (Ea) and let θ = (θ1, . . . , θ10), Θ = Pθ, Φ = Θ−1. 42 B. Doubrov, Y. Machida and T. Morimoto Note that Θ and Φ are GL(V ) (= GL(10,R))-valued local functions on M . Similarly as in Example 4.7 we may regard Φ as giving a cross-section Q(0) Q(0) ι−−−−→ GL(V ) Φ x xΘ−1 y M M ϕ−−−−→ Flag(V, φ), where ϕ is an embedding corresponding to (Ea) and Q(0) the associated extrinsic bundle (recall Section 5.3). Therefore, the gl(V )-valued form Ω given by Ω = Φ−1 dΦ = −dΘ Θ−1 is the pull-back to M of the canonical form ω(0) on Q(0). Write Ω = F0ω0 + F1ω1 + F2ω2, where {ω0, ω1, ω2} is the coframe dual to {Z,X, Y }. By using the relations of (Ea) we can easily write down {Fi} and see that F0 and F2 do not depend on the parameter a and F1 = aE74 + ◦ F 1, where E74 is the matrix element of (7, 4) and ◦ F 1 does not depend on a. Recalling the discussion of Section 5, we see that the 1-st order component χ(0)1 of the structure function χ(0) of Q(0) is then represented by γ = E74 ⊗A∗8. Now take a closer look at the complex 0→ ḡ1 ∂−→ Hom (g−, gl(V )/ḡ)1 ∂−→ Hom ( ∧2g−, gl(V )/ḡ ) 1 . If we write down explicitly the adjoint representation of sp(4,R) with respect to the basis {A1, A2, . . . , A10}, it is not difficult to verify that γ is a cocycle but not a coboundary. Hence it defines a non-trivial cohomology in H1 1 (g−, gl(V )/ḡ). It then proves, by virtue of corollary to Theorem 5.5, that (Ea) is not isomorphic to (E0) for any a 6= 0. � It should be remarked that γ = E74⊗A∗8 has weight α1−2α2 and in turn gives a lowest weight vector of H1 (g−, gl(V )/ḡ). Note also that systems (Ea1) and (Ea2) for different non-vanishing a1 and a2 are equivalent to each other. Cases (i) of the contact A2 geometry was already treated in Example 4.9 of Section 4. In par- ticular, we have the standard model given by the Segre embedding P (U)×̆P (U∗)→ P ( U⊗̆U∗ ) , where dimU = 3. The corresponding model equation can be locally expressed in the following form of a system of (weighted) second order differential equations{ X2u = 0, Y 2u = 0. Similar to Proposition 6.8 we can show that the solution space of the system{ X2u = aY u, Y 2u = 0, Extrinsic Geometry and Linear Differential Equations 43 is 8-dimensional and provides a non-trivial deformation of the Segre embedding for any non-zero constant a. Finally, let us describe the standard model for case (iii) for the contact G2 geometry. Let U be the 7-dimensional standard representation of g. It is easy to check that the g-module Λ2U decomposes as V ⊕ U , where, as above, V = g is the adjoint representation. The projection to the first summand gives us g-invariant map Λ2U → V , while the projection to the second summand defines an anticommutative multiplication on U . The standard embedding ϕmodel is generated in this case by the following projective em- bedding ϕ4 model : G/G0 = N0 Gr2(U) −→ P 13 = P (V ), where N0 Gr2(U) is the Grassmann manifold of all 2-dimensional null subalgebras in U . We may call this map the contact sub-Plücker embedding. Using Proposition 6.7 we see that the cohomology describing the fundamental invariants of osculating maps ϕ : (M, f)→ L/L0 ⊂ Flag(V, φ) is trivial in this case. So the standard model embedding ϕmodel is rigid. We remark that the first cohomology H1(g−, V ) = ⊕ H1 r (g−, V ) represents a system of differ- ential equations, and V = ⊕Vq represents its solution space, see [24]. In our case the computation shows that H1(g−, V ) = H1 0 (g−, V ) ∼= Hom0 ( S2g−1, V0 ) . Here Hom0 ( S2g−1, V0 ) is the 7-dimensional irreducible g0-submodule of Hom ( S2g−1, V0 ) . Exp- licitly, we get the system of (weighted) linear differential equations on M which is G2-invariant and has 14-dimensional solution space defining the standard embedding Z2u = W 2u = Y Zu = XWu = 0,( 2ZX + Y 2 ) u = ( 2YW +X2 ) u = 0, (XY + Y X +WZ + ZW )u = 0, where we take the local coordinates (x, y, z, w, v) in X such that the contact distribution is represented by {dv − 2ydx+ xdy + zdw = 0}, Z = ∂ ∂z , W = ∂ ∂w − z ∂ ∂v , X = ∂ ∂x + 2y ∂ ∂v , Y = ∂ ∂y − x ∂ ∂v . See [13] for more details. Here u is an unknown function u = u(x, y, z, w, v) on X. 6.6 Realizability of deformations Proposition 6.6 lists all possible types of non-rigid rational homogeneous varieties of simple Lie groups. However, it does not provide explicit examples when they are really rigid. For quadrics one can take the standard representation of Bl, Dl or A3 ∼= D3, as a non- degenerate quadric in Pn is clearly non-rigid. In case of projective spaces one can consider, for example, the osculating map ϕ : Pn → Flag ( R(n+1)(n+2)/2, φ ) generated by the Veronese embedding ν2 : Pn → P (n+1)(n+2)/2−1, which corresponds to the S2(V ) representation of SL(V ). The filtration φ has type ((n+ 1)(n+ 2)/2, n+ 1, 1) in this case. Let (x1, . . . , xn) be affine coordinates on Pn. In these coordinates the Veronese embedding is specified by the space of all polynomials of degree ≤ 2 in x1, . . . , xn. The corresponding PSL(n+ 1,R) invariant system of PDEs ∂3u ∂xi∂xj∂xk = 0. 44 B. Doubrov, Y. Machida and T. Morimoto We can provide an explicit deformation of the Veronese embedding by replacing the equation ∂3u (∂x1)3 = 0 with ∂3u (∂x1)3 = ∂2u (∂x2)2 . It is easy to see that this deformed system is still compatible and has the solution space of dimen- sion (n+ 1)(n+ 2)/2, thus providing an explicit deformation of the Veronese embedding. Com- puter algebra calculations show that the dimension of the symmetry algebra of this system is less than the one of the trivial system implying that this deformation is non-trivial. Consider now the adjoint variety of sp(2n,R). It was explicitly demonstrated in the previous subsection that it is non-rigid for the case of sp(4,R). Assume now that n is arbitrary. Let V = sp(2n,R) be the adjoint representation. Its hig- hest weight is 2ω1, and ḡ = csp(2n,R). We have the following decompositions of gl(V ) and ḡ⊥ ⊂ gl(V ): gl(V ) = V4ω1 ⊕ V2ω1+ω2 ⊕ V2ω2 ⊕ V2ω1 ⊕ Vω2 ⊕ V0, ḡ⊥ = V4ω1 ⊕ V2ω1+ω2 ⊕ V2ω2 ⊕ Vω2 . In particular, H1 +(g−, V4ω1) does not vanish, which suggests that the adjoint variety of sp(2n,R) is non-rigid for any n ≥ 3. Consider now A-type contact case in more detail. It corresponds to (Al, {α1, αl}). Its minimal embedding is given by the highest root orbit of the adjoint representation of SL(l + 1,R). However, it can be shown that this orbit is non-rigid only for l = 2. Indeed, we have the following decomposition of ḡ⊥, V = sl(l + 1,R) depending on l: ḡ⊥ = V2ω1+2ωl ⊕ V2ω1+ωl−1 ⊕ Vω2+2ωl ⊕ Vω2+ωl−1 ⊕ Vω1+ωl , l ≥ 4, ḡ⊥ = V2ω1+2ω3 ⊕ V2ω1+ω2 ⊕ Vω2+2ω3 ⊕ V2ω2 ⊕ Vω1+ω3 , l = 3, ḡ⊥ = V2ω1+2ω2 ⊕ V3ω1 ⊕ V3ω2 ⊕ Vω1+ω2 , l = 2. In particular, we see that for l ≥ 3 this decomposition has no components U , for which the cohomology H1 +(g−, U) is non-trivial (see Proposition 6.7). Lemma 6.9. Consider the injective map sl(l+ 1,R)→ gl(V ), l ≥ 2, where V is the irreducible representation with the highest weight ρ = ω1 + ω2 + · · ·+ ωl. Then the decomposition of gl(V ) contains the components V(l+1)ω1 and V(l+1)ωl with multiplicity 1. Proof. Indeed, the multiplicity of V(l+1)ω1 and V(l+1)ωl in the decomposition of gl(Vρ) can be computed using the Littlewood–Richardson rule [9]. To do this, we note that the representa- tion Vρ is self-dual, and, hence gl(Vρ) is isomorphic to Vρ ⊗ Vρ. In terms of Young tableaux the representation Vρ corresponds to the tableau λ of shape (l, l− 1, . . . , 1) (l boxes in the first row, l − 1 boxes in the second row, . . . , 1 box in the last row). As V(l+1)ω1 and V(l+1)ωl are dual to each other, it is clear that their multiplicities in the decomposition of Vρ ⊗ Vρ are the same. So, we can concentrate on computing the multiplicity of V(l+1)ω1 , which corresponds to any tableau of shape (k + l+ 1, k, . . . , k) (l+ 1 rows in total). To fix k, we have to make sure that the tableau has exactly l(l+ 1) (twice the number of boxes in the tableau of Vρ). So, k = l − 1 and we should represent V(l+1)ω1 by the tableau ν of shape (2l, l − 1, . . . , l − 1) (l + 1 rows). Finally, applying the Littlewood–Richardson rule, the multiplicity of V(l+1)ω1 is equal to the number of Littlewood–Richardson tableaux of shape ν/λ and of weight λ. It is easy to see that the first row of ν/λ should be filled by 1. The next empty box appears in the 3rd row, Extrinsic Geometry and Linear Differential Equations 45 and it should be filled by 2 (otherwise we would not be able to fill in the 3rd column by the strictly increasing sequence). In the same way, it is easy to see that the 4th row should be filled by 2 and 3 and so on. So, we have a unique Littlewood–Richardson tableaux of shape ν/λ and of weight λ. See, for example, the picture below for l = 4: • • • • 1 1 1 1 • • • • • 2 • 2 3 2 3 4 � So, according to Proposition 6.7 the cohomology H1 + ( g−, ḡ ⊥) does not vanish in this case. This leads to the conjecture that a rational homogeneous variety corresponding to the represen- tation Vρ is non-rigid. 7 Equivalence problem of extrinsic geometries – general case 7.1 Generalizations to the non-principal and to the infinite-dimensional cases In this section we extend the discussions in Section 5 to obtain the invariants of extrinsic geo- metries in more general settings. First we drop the assumption that L/L0 is embedded in a flag variety Flag(V, φ) since it is not the representation itself of L on V but the filtered structure on L/L0 that matters. Second we will not assume the conditions of G 0 -invariance (C1) and (C2) which were needed for “principal reduction” to obtain extrinsic Cartan connections. Here we perform rather “step reduction” which does not necessarily lead to Cartan connections, but still gives a general algorithm to obtain the complete invariants of an extrinsic geometry. Third we extend our discussions to the cases when L may be infinite dimensional. This generalization will have interesting applications to extrinsic geometries of non-linear differential equations with respect to infinite-dimensional Lie groups such as the contact transformation groups. To generalize to infinite-dimensional transformation groups, we rather treat infinitesimal transformations. Definition 7.1 ([21, 32]). A transitive Lie algebra sheaf (of infinitesimal transformations) on a filtered manifold (N, f) is a subsheaf L (of Lie algebras) of the Lie algebra sheaf TN satisfying: (i) L leaves invariant f, that is, [L, fp] ⊂ fp for all p ∈ Z, (ii) L is transitive, that is, the evaluation map Lx 3 [X]x 7→ Xx ∈ TxN is surjective for all x ∈ N , (iii) L is defined by an involutive system of differential equations. Note that the condition (iii) means that L is “continuous” in the sense of Lie and Cartan. For the notion of involutive differential equations on filtered manifolds, see [23]. Let P(L) denote the pseudo-group of local transformations of (N, f) generated by L, namely by all local flows ft of all sections X of L. Let ϕ1 : (M1, fM1) → (N, fN ) and ϕ2 : (M2, fM2) → (N, fN ) be two morphisms of filtered manifolds, and let x1 ∈ M1, x2 ∈ M2. We say that the germ of ϕ1 at x1 is L-equivalent to the germ of ϕ2 at x2, if there exist a local isomorphism h : (M1, fM1) → (M2, fM2) with h(x1) = x2 and a ∈ P(L) such that a ◦ ϕ1 = ϕ2 ◦ h. It is under this L-equivalence that we are going to study the equivalence problem. Before going further, let us recall some basic facts about our L. 46 B. Doubrov, Y. Machida and T. Morimoto First of all we note that each stalk Lx has a natural filtration {Lpx}p∈Z induced from that of (N, fN ) as defined by Lpx = {{ [X]x ∈ Lx \| Xx ∈ fpx } (p ≤ 0),{ ξ ∈ Lp−1 x \| [ξ,Lix] ⊂ Li+px for all i < 0 } (p ≥ 1). Then it is easy to verify that [Lpx,Lqx] ⊂ Lp+qx . Thus, ( Lx, {Lpx} ) is a filtered Lie algebra, and passing to the quotient we get a graded Lie algebra grLx. As shown later, gr fx is anti-isomorphic to gr− Lx ( = ∑ p<0 grp Lx ) . With the notion of weighted jet bundle the filtration {Lpx} can be defined alternatively as follows. Regarding TN as a filtered vector bundle over a filtered manifold (N, fN ), let Ĵ (k)TN be the weighted k-th jet bundle. Denote by ĵ (k) x the canonical map TN x → Ĵ (k)TN x for x ∈ N . Then we have Lp+1 x = { [X]x ∈ Lx \| ĵ(p) x X = 0 } . Now let L(k) x = { ĵ(k) x X \| [X]x ∈ Lx } = Lx/Lk+1 x , L(k) = ⋃ x∈N L(k) x . Then L(k) ⊂ Ĵ (k)TN may be regarded as a system of differential equations defining L. The condition (iii) of the definition of Lie algebra sheaf may be understood to mean that there exists k0 such that L(k0) is weighted involutive and L(l) is the prolongation of L(k0) for l > k0. From the formal theory of differential equations, it is equivalent to saying that L(k) are vector bundles for all k. Taking the projective limit, we set L(∞) x = lim ←− L(k) x , which carries the structure of filtered Lie algebra and is called the formal algebra of L at x. Note that from the transitivity of L it follows that L(∞) x and L(∞) y are isomorphic for all x, y in a connected component of N . Assuming N connected if necessary, we fix a filtered Lie algebra l, which is anti-isomorphic to all L(∞) x , x ∈ N . A proper generalization of the bundle L → L/L0 to infinite dimension may be to introduce the following principal bundle equipped with a Pfaff system: Theorem 7.2. There is associated to a transitive Lie algebra sheaf L on a filtered manifold (N, f) a principal fibre bundle L over N with structure group L0 and l-valued 1-form ωL on L such that (i) L0 is a possibly infinite-dimensional Lie group equipped with a filtration {Lp}p≥0 consisting of closed normal subgroups of L0 with the Lie algebra of Lp being lp and L0 = L(∞) = lim ←− L(k), L = L(∞) = lim ←− L(k), where L(k) = L0/Lk+1, L(k) = L/Lk+1. Extrinsic Geometry and Linear Differential Equations 47 (ii) The map ( ωL ) z : TzL→ l is a filtration preserving isomorphism for all z ∈ L. (iii) 〈ωL, Ã〉 = A for all A ∈ l0. (iv) The adjoint action of L0 on l0 is extended to l and R∗aωL = Ad(a)−1ωL for all a ∈ L0. (v) dωL + 1/2[ωL, ωL] = 0. (vi) If X is a section of L, then there is a unique lift X̂ of X to L such that L X̂ ωL = 0. Conversely, if a local vector field Z on L satisfies LZωL = 0, then there exists a section X of L such that Z = X̂. Note that L and L0 can be infinite dimensional. But since they are projective limits of finite-dimensional objects, we can treat those infinite-dimensional objects similarly to the finite- dimensional case. Proof. To prove the theorem we employ the scheme developed in [22]. First we note that since L is transitive on N , (N, fN ) has a constant symbol, say m. Let R(k)(N, fN ,m) or simply R(k) be the reduced frame bundle of order (k+ 1) and set R = R(∞) = lim ←− R(k) (see [22, p. 316]). Recall that every automorphism h of (N, fN ) is uniquely lifted to an automorphism ĥ(k) of R(k). Therefore we have a lift P̂(L) (k) of P(L) to R(k). Passing to the infinitesimal, every infinitesimal automorphism X of (N, fN ) is uniquely lifted to the one X̂(k) on R(k) and the Lie algebra sheaf L to the one L̂(k) on R(k). Clearly, we have P ( L̂(k) ) = P̂(L) (k) . By evaluation, L̂(k) defines a subbundle D(k) ⊂ TR(k), which is, as easily seen, of constant rank and completely integrable. Now choose z̊(k) ∈ R(k) (k = 0, 1, . . . ) such that πk+1 k ( z̊(k+1) ) = z̊(k), where πk+1 k denotes the projection R(k+1) → R(k). Let L(k) be the maximal integral manifold of D(k) through z̊(k), and set L = lim ←− L(k), z̊ = lim ←− z̊(k). Then we see immediately that L(k+1) → L(k) is a surjective submersion for all ∞ ≥ k ≥ −1, where we set L−1 = N . Not only this, we are now going to see that L(k) → L(j) are all principal fibre bundles for k ≥ j. Let ωR be the canonical 1-form on R, which is a 1-form on R taking values in E(m) = m⊕g0(m), where G0(m) is the structure group of R→ N and g0(m) is the Lie algebra of G0(m). Let ωL be the restriction of ωR to L: ωL = ι∗ωR, where ι denotes the inclusion L → R. If Φ ∈ P̂(L), then Φ∗ωR = ωR and Φ(L) = L. Therefore Φ∗ωL = ωL. It then follows that there is a subspace EL ⊂ E(m) such that we have an isomorphism( ωL)z : TzL→ EL for all z ∈ L, and we have dωL + 1 2 γL(ωL, ωL) = 0 with a constant structure function γL ∈ Hom ( ∧2EL, EL ) , because P̂(L) is transitive on L. We see therefore that γL : EL × EL → EL defines a Lie algebra structure on EL. Moreover, if we set g0(L) = g0(m) ∩ EL, the Lie algebra structure defined by γL\|g0(L)×g0(L) coincides with the one induced from g0(m). 48 B. Doubrov, Y. Machida and T. Morimoto Let X,Y ∈ L. Substituting X̂, Ŷ into the structure equation, we have X̂ωL ( Ŷ ) − Ŷ ωL ( X̂ ) − ωL ([ X̂, Ŷ ]) + γL ( ωL ( X̂ ) , ωL ( Ŷ )) = 0. But since LX̂ωL = 0, we have X̂ω ( Ŷ ) = ( LX̂ωL )( Ŷ ) + ωL(LX̂ Ŷ ) = ωL ([ X̂, Ŷ ]) . Therefore we have ωL ([ X̂, Ŷ ]) + γL ( ωL ( X̂ ) , ωL ( Ŷ )) = 0, which implies that the map Lx −̂→ L̂z (ωL)z−−−→ EL (z ∈ L, x = πN (z) ∈ N) gives an anti-isomorphism L(∞) x ω∧x−−→ (EL, γL) of Lie algebras, where L(∞) x denotes the formal algebra of L at x. Recalling that EL and L(∞) x have natural filtrations { φpEL } and { φpL(∞) x } , we remark that ω∧x ( φpL(∞) x ) = φpEL for all p ∈ Z. In fact, it is easy to see that it holds for p ≤ 0. But in both filtrations the space φp (p ≥ 0) is determined by φp = { X ∈ φp−1 \| [X,φa] ⊂ φp+a for all a < 0 } . Therefore the assertion is valid for all p ∈ Z. In particular, we have 1. gr fN,x is anti-isomorphic to gr− Lx for x ∈ N . 2. For X ∈ Lx, z ∈ L with πN (z) = x, and k ∈ Z, jkxX = 0 if and only if X̂z ≡ 0 mod φk+1TzL. In relation to (2), in general, we can prove that if X is an infinitesimal automorphism of a filtered manifold (M, f), then, for k < 0, jkxX = 0 if and only if Xk ∈ fk+1 x . Let us show that if za, zb ∈ L for z ∈ L, a, b ∈ G0(m), then zab ∈ L. In fact, since z, za ∈ L and P̂(L) is transitive on L, there exist neighborhoods U , U ′ of x = πR(z) in N and a diffeomorphism Φ: ( πR )−1 (U) → ( πR )−1 (U ′) such that Φ∗ωR = ωR, Φ(wg) = Φ(w)g, w ∈( πR )−1 (U), g ∈ G0(m), Φ (( πR )−1 (U)∩L ) ⊂ ( πR )−1 (U ′)∩L and Φ(z) = za, where πR denotes the projection R → N . Then since zb ∈ L ∩ ( πR )−1 (U), we have zab = Φ(z)b = Φ(zb) ∈ L, which proves the assertion. Now it is easy to see that L is a principal fibre bundle over N with structure group G0(L), where G0(L) is a Lie subgroup of G0(m) with Lie algebra g0(L), which can be identified with l0. It is not difficult to verify that all the statements in Theorem 7.2 hold. � Remark 7.3. The statement (vi) of Theorem 7.2 means that (L, ωL) can be regarded as a defin- ing equation of L, in other words, an infinitesimal automorphism X of (N, f) is a section of L if and only if the lift X̂ of X to R is tangent to L, which is, in turn, equivalent to saying that the lift X̂(k0) of X to R(k0) is tangent to L(k0) for certain integer k0. This integer k0 is given by the condition H1 r (gr− l, gr l) = 0 for r ≥ k0. The existence of such k0 is assured by the finiteness of this cohomology group (see [21]). Extrinsic Geometry and Linear Differential Equations 49 It being prepared, let ϕ : (M, fM ) → (N, fN ) be a morphism. Then via pull-back we obtain the principal fibre bundle ϕ∗L over M with the structure group L0 endowed with an l-valued 1- form ϕ̃∗ω, where ϕ̃ : ϕ∗L→ L is a canonical inclusion. The bundle with Pfaff forms (ϕ∗L, ϕ̃∗ω), denoted also by (Qϕ, ωϕ), is called an extrinsic bundle associated with ϕ. From now on we assume grϕ∗,x : gr(fM )x → gr(fN )ϕ(x) is injective for all x ∈ M , so that grϕ∗,x is an injective graded Lie algebra homomorphism. Let g− = ⊕ p<0 gp be a graded subalgebra of gr l− = ⊕ p<0 l. Definition 7.4. We say that ϕ : (M, fM )→ (N, fN ) has constant symbol of type (g−,L), if for any x, y ∈ M there exist a ∈ P(L) and isomorphisms of graded Lie algebras βx and βy which make the following diagram commutative: gr(fM )x gr(fN )ϕ(x) g− gr(fM )y gr(fN )ϕ(y). ϕ∗,x gr a βx βy ϕ∗,x Then we have Proposition 7.5. The map ϕ : (M, fM ) → (N, fN ) is of type (g−,L), if and only if (Qϕ, ωϕ) satisfies the following criterion: For any x ∈M there exists z ∈ Qϕ over x such that grωz(gr(fM )x) = g−. Now we are going to construct the invariants of extrinsic geometries ϕ : (M, fM ) → (N, fN ) under L. To avoid surplus complexity, we assume: (C0) l = ⊕̂ lp, that is, l is isomorphic to the completion of gr l obtained as a projective limit of partial sums ⊕ p≤k lp for k ≥ 0. We also restrict ourselves to morphisms ϕ of constant symbol, say of type (g−,L). We follow the discussions in Section 5 by making necessary modifications to adapt to this general case. Let ḡ = Prol(g−, l) be the relative prolongation of g− in l. Similarly as before we define G0 = { a ∈ L(0) \| Ad(a)g− = g− } and G(k), ḡ(k) as well as their filtrations { G(k)p } , { ḡ(k)p } . Recall thus for instance, ḡ(k) = ḡ(k)0 = ⊕ 0≤p≤k ḡp ⊕ ⊕ q>k lq, ḡ(k)k = ḡk ⊕ ⊕ q>k lq. We choose complementary subspaces ḡ′ = ⊕ g′p to ḡ in l so that lp = gp ⊕ g′p, and we have a direct sum decomposition l = E(k)⊕ E′(k) with E(k) = g− ⊕ ḡ(k), E′(k) = ⊕ q≤k g′q. 50 B. Doubrov, Y. Machida and T. Morimoto We choose also complementary subspace W = ⊕ p≥1Wp such that we have for all p ≥ 1 Hom (g−, l/ḡ)p = Wp ⊕ ∂(l/ḡ)p. Let (Qϕ, ωϕ) be the original extrinsic bundle induced from ϕ : (M, fM )→ (N, fN ). First we define (Q(0), ω(0)) by Q(0) = { z ∈ Qϕ \| gr(ωϕ)z ( gr(fM )x ) = g− } and ω(0) = ι∗ωϕ, where ι : Q(0)→ Qϕ is the canonical inclusion. Now we construct inductively Q(k)→ B(k − 1), ω(k), χ(k), in such a way that: (i) Q(k) is a principal fibre bundle over the base space B(k − 1) = Q(k − 1)(k−1) = Q(k − 1)/Ḡ(k)k with structure group G(k)k. Furthermore there is a canonical inclusion ι : Q(k)→ Q(k−1) as principal fibre bundles over B(k − 1), (ii) ω(k) = ι∗ω(k − 1) and R∗aω(k) = Ad(a)−1ω(k) for a ∈ Ḡ(k)k, (iii) If we write ω(k)II = χ(k)ωI according to the direct sum decomposition l = E(k)⊕E′(k), then ω(k)I : TzQ(k)→ E(k) is a filtration preserving linear isomorphism for all z ∈ Q(k), (iv) Define a map χ(k) : Q(k) → Hom(E(k), E′(k)) by ω(k)II = χ(k)ω(k)I and decompose it as χ(k) = χ−(k) + χ+(k) and χ(k)p = χ−(k)p + χ+(k)p by requiring that χ−(k)p : Q(k)→ Hom(g−, E ′(k))p, χ+(k)p : Q(k)→ Hom ( ⊕ 0≤i≤k ḡi, E ′(k) ) p , χ(k) = ∑ χ(k)p, χ(k)− = ∑ χ(k)−p , χ(k)+ = ∑ χ(k)+ p . Then (a) χ(k)i = ι∗χ(k − 1)i for i < k, (b) χ(k)p = 0 for p ≤ 0, (c) χ−(k)k ∈Wk. Extrinsic Geometry and Linear Differential Equations 51 L Qϕ Q(0) Q(1) Q(2) · · · Q(k) Q(k+1) ... ... ... ... ... Q(k)(k+1) Q(k + 1)(k+1) B(k+1) Q(k)(k) B(k) ... ... B(k − 1) ... Q(1)(2) Q(2)(2) B(2) ... ... Q(0)(1) Q(1)(1) B(1) L(0) Q (0) ϕ Q(0)(0) B(0) N M M Gk+1 Gk G2 G1 G0 ϕ The normalization procedure is illustrated by the above diagram. Note the main differences from the previous case are first of all that Q(k) is no more a principal bundle over M but only over Q(k − 1)(k−1) and second that χ+(k) here does not vanish, in general. The construction is based on the following formula: Proposition 7.6. R∗expAk+1 χ−(k)k+1 = χ−(k)k+1 − ∂Ak+1 for Ak+1 ∈ ḡ(k)k+1(= lk+1). Proof. We have already shown this formula in Section 5 under the assumptions of Ḡ0 invariance conditions (C1), (C2), but to prove it in our general case we should argue more carefully. In the formula ∂ denotes the coboundary operator in the complex 0→ l/ḡ ∂→ Hom(g−, l/ḡ)→ · · · , identifying ḡ′ with l/ḡ, and writing simply ∂ for ∂ ◦π, where π is the projection: l→ l/ḡ. During the proof, we simply write ω(k) = ω, χ(k) = χ = ∑ i≥1 ( χ+ i + χ−i ) , α = expAk+1. Recall that R∗αω = Ad ( α−1 ) ω, ω = ωI + ωII , ωII = χωI . 52 B. Doubrov, Y. Machida and T. Morimoto Then we have R∗α(ωI + ωII) = Ad ( α−1 ) (ωI + ωII) = Ad ( α−1 )I I ωI + Ad ( α−1 )II I ωII + Ad ( α−1 )I II ωI + Ad ( α−1 )II II ωII = Ad ( α−1 )I I ωI + Ad ( α−1 )II I χωI + Ad ( α−1 )I II ωI + Ad ( α−1 )II II χωI = ( Ad ( α−1 )I I + Ad ( α−1 )II I χ ) ωI + ( Ad ( α−1 )I II + Ad ( α−1 )II II χ ) ωI , where we use the notation Ad ( α−1 )I II to denote the Hom(E(k), E′(k))-component of Ad ( α−1 ) and similar notations. Therefore we have R∗αωI = ( Ad ( α−1 )I I + Ad ( α−1 )II I χ ) ωI , R∗αωII = ( Ad ( α−1 )I II + Ad ( α−1 )II II χ ) ωI . On the other hand, we have R∗αωII = (R∗αχ)(R∗αωI). Therefore we have( Ad ( α−1 )I II + Ad ( α−1 )II II χ ) ωI = (R∗αχ) ( Ad ( α−1 )I I + Ad ( α−1 )II I χ ) ωI . Letting ωI take values in g−, we have Ad ( α−1 )I− II + Ad ( α−1 )II II χ− = (R∗αχ) ( Ad ( α−1 )I− I + Ad ( α−1 )II I χ− ) , and then Ad ( α−1 )I− II + Ad ( α−1 )II II χ− = (R∗αχ −) ( Ad ( α−1 )I− I− + Ad ( α−1 )II I− χ− ) + (R∗αχ +) ( Ad ( α−1 )I− I+ + Ad ( α−1 )II I+ χ− ) , where Ad ( α−1 )I− I− denotes the Hom(g−, g−)-component of Ad ( α−1 ) and the same convention for the similar notation. Now enter vp ∈ gp (p < 0) into the above formula and catch the (l/ḡ)i+p -component for i ≤ k + 1. Taking into account that Ad ( α−1 )I− II = − ad(Ak+1)I − II + higher order terms, Ad ( α−1 )II II = idII − ad(Ak+1)IIII + higher order terms and that( Ad ( α−1 )I− I+ + Ad ( α−1 )II I+ χ− ) (vp) ∈ lk+1+p + lk+2+p + · · · and therefore( R∗αχ + j+p ) ( Ad ( α−1 )I− I+ + Ad ( α−1 )II I+ χ− ) (vp) = 0 for j ≤ k + 1, we have R∗αχ − i = χ−i for i ≤ k R∗αχ − k+1 = χ−k+1 − ∂Ak+1, which proves the proposition. � Extrinsic Geometry and Linear Differential Equations 53 We can then easily carry out the inductive construction of Q(k), and by passing to limit we obtain Q = Q(∞), ω = ω(∞), χ = χ(∞). The structure function χ : Q→ Hom(ḡ, l/ḡ) is an invariant of Q and that of the initial morphism ϕ : (M, f)→ (N, fN ). To examine it more closely, we write χ = ∑( χ−k + χ+ k ) , χ+ k = ∑ 0≤i<k χ+i k , where χ+i k takes its value in Hom(ḡi, ḡk). Then we have Proposition 7.7. ∂χ−k+1 = Ψ−k+1 (χ`, Dχ`; ` ≤ k), χ+ k+1 = Ψ+ k+1 ( χ−` , χ + m, Dχ − ` , Dχ + m; ` ≤ k + 1, m ≤ k ) , where Ψ−k+1 is a vector valued function determined explicitly only by χ` (` ≤ k) and their derivatives, and Ψ+ k+1 is determined only by χ−` (` ≤ k + 1), χ+ m (m ≤ k) and their derivatives. Moreover Ψ−k+1 = 0 if χ` = 0 (` ≤ k), and Ψ+ k+1 = 0 if χ` = 0 (` ≤ k) and χ−k+1 = 0. Proof. From the structure equation dω + 1 2 [ω, ω] = 0, ω = ωI + ωII , ωII = χωI , it follows that dωI + 1 2 [ωI , ωI ] + [ωI , ωII ]I + 1 2 [ωII , ωII ]I = 0, dωII + [ωI , ωII ]II + 1 2 [ωII , ωII ]II = 0, and then dωI + 1 2 [ωI , ωI ] + [ωI , χωI ]I + 1 2 [χωI , χωI ]I = 0, d(χωI) + [ωI , χωI ]II + 1 2 [χωI , χωI ]II = 0, which yields [ωI , χωI ]II − χ ( 1 2 [ωI , ωI ] ) − χ([ωI , χωI ]I) + 1 2 [χωI , χωI ]II − χ ( 1 2 [χωI , χωI ]I ) + dχ ∧ ωI = 0. Now evaluating the above equation at ṽp∧w̃q, where ṽ, w̃ denote the tangent vectors such that ωI(ṽp) = vp ∈ gp, ωI(w̃q) = wq ∈ gq and p, q < 0, and looking at the (l/ḡ)k+1+p+q-component, we have (∂χ−k+1)(vp, wq) = A ∑ a+b=k+1, a,b>0 ( χ−a [vp, χ − b (wq)]I + [χ−a (vp), χ − b (wq)]II ) +A ∑ a+b+c=k+1, a,b,c>0 χ−a [χ−b (vp), χ − c (wq)]I − ṽpχ−k+1+p(wq) + w̃qχ − k+1+q(vp) +A ∑ 0<b<k+1 χ+ k+1+p+q[vp, χ − b (wq)]I +A ∑ b+c<k+1, b,c>0 χ+ k+1+p+q[χ − b (vp), χ − c (wq)]I , 54 B. Doubrov, Y. Machida and T. Morimoto where A denotes the alternating sum in vp, wq. The last formula defines explicitly the func- tion Ψ−k+1. Next we evaluate the previous formula at Ãi ∧ ṽp, where Ai ∈ ḡi (i ≥ 0), vp ∈ gp (p < 0) and Ãi, ṽp are the corresponding tangent vectors. Looking at (l/ḡ)k+1+p-component, we have[ vp, χ + k+1Ai ] II = [ Ai, χ − k+1−ivp ] II − χ−k+1−i [Ai, vp]− χ+ k+1+p [Ai, vp] − ∑ a+b+i=k+1 χ−a [ Ai, χ − b vp ] I − ∑ b+i<k+1 χ+ k+1+p [ Ai, χ − b vp ] I + ∑ a+b=k+1 χ−a [ vp, χ + b Ai ] I + ∑ b<k+1 χ+ k+1+p [ vp, χ + b Ai ] I + ∑ a+b=k+1 [ χ+ a Ai, χ − b vp ] II − ∑ a+b+c=k+1 χ−a [ χ+ b Ai, χ − c vp ] I − ∑ b+c<k+1 [ χ+ b Ai, χ − c vp ] I + Ãiχ − k+1(vp)− ṽpχ+ k+1+p(Ai), where a, b, c > 0. Since ∂ : l/ḡ → Hom(g−, l/ḡ) is injective, the above formula determi- nes Ψ+ k+1. � Note that we know the finitness of the cohomology group: Proposition 7.8. There exists an integer r0 such that H i r(g−, l/ḡ) = 0 for all r > r0 and all i. Proof. The proof of Theorem 2.1 in [21] applies also to this case. � Now we have Theorem 7.9. Let ϕ : (M, fM )→ (N, fN ) be a morphism of type (g−,L). Then we can construct the series of principal fibre bundles Q(k)→ B(k−1) with l-valued 1-forms ω(k) and the structure functions χ(k)(k) : B(k) → W (k) for k = 0, 1, 2, . . . . And χ(k0)(k0) fulfills the complete set of invariants of ϕ, where k0 is an integer such that H1 k(g−, l/ḡ) = 0 for all k > k0. Let G− (resp. L−) be the Lie group with the Lie algebra g− (resp. l−). Then we have local embeddings uniquely defined up to L-equivalence: ϕmodel : G− → L− → (N, fN ), which we call the standard embedding. Corollary. Let ϕ, χ(k0)(k0) be as in the preceding theorem. If χ(k0)(k0) = 0 identically, in par- ticular, if H1 r (g−, l/ḡ) = 0 for r > 0, then ϕ is formally L-equivalent to the standard embedding at every point of M . In fact, if the hypothesis of the corollary holds, then the differential equation for an L- equivalence between ϕ and ϕmodel is proved to be formally integrable and involutive, and there- fore has a formal solution. Moreover, it has a local analytic solution in the analytic category. If l is finite-dimensional, then the differential equation is of finite type and has a local smooth solution even in the C∞-category. Extrinsic Geometry and Linear Differential Equations 55 7.2 Examples Example 7.10. As an example, consider the embeddings ϕ : (M, fM )→ (N, fN ), where (N, fN ) is a contact manifold of dimension 2n + 1. In other words, (N, fN ) is a filtered manifold of constant type m = m−2 ⊕ m−1 = n2n+1, where n2n+1 is a (2n + 1)-dimensional Heisenberg Lie algebra. Note that the Lie bracket ∧2m−1 → m−2 defines a symplectic form on m−1 up to a non-zero scale. In this case, l is the (infinite-dimensional) graded Lie algebra associated with the filtered Lie algebra of all contact vector fields on N . Let (xi, y, zi) be a local coordinate system on N such that the contact distribution f−1 N is defined as an annihilator of the 1-form θ = dy − z1 dx1 − · · · − zn dxn. Then it is well-known that all contact vector fields Xf are parametrized by a single function f on N such that θ(Xf ) = f, (LXf θ) ∧ θ = 0. The Lie bracket of contact vector fields induces the bracket {f, g} of functions f , g on N such that X{f,g} = [Xf , Xg]. Explicitly, we have: {f, g} = f ∂g ∂y − g∂f ∂y + n∑ i=1 ( df dxi ∂g ∂zi − dg dxi ∂f ∂zi ) , where d dxi = ∂ ∂xi + zi ∂ ∂y for all i = 1, . . . , n. The Lie algebra l can be described as a set of all contact vector fields Xf , where f runs through all polynomials in (xi, y, zi), 1 ≤ i ≤ n. In the following, we identify l with the space of such polynomials equipped with the above Lie bracket. The Lie algebra m can be identified with l−, which corresponds to the subspace 〈1, xi, zi \| 1 ≤ i ≤ n〉. Let us classify all graded subalgebras g− of m up to Aut0(m). • Case 1: g− = g−1 ⊂ m. Then g− is necessarily commutative and thus, is an isotropic subspace of m−1. As the symplectic group acts transitively on isotropic subspaces of the same dimension in m−1, we see that the g− is uniquely determined up to Aut0(m) (=the grading preserving group of automorphisms of m) by its dimension. • Case 2: g− is 2-graded. Then we have g−2 = m−2. Assuming that g− is generated by g−1, we get that g−1 can be an arbitrary non-isotropic subspace of m−1. Again, modulo the action of Aut0(m) such subspaces are uniquely determined by its dimension and the dimension of the kernel of the restriction of the symplectic form to g−1. Let us compute Prol(g−, l) in cases, when g− is maximal isotropic in m−1 or when g− is two-graded, and the restriction of the symplectic form to g−1 is non-degenerate. • Case 1: g− is a maximal isotropic subspace in l−1. Then up to Aut0(m), we can assume that g− = 〈z1, . . . , zn〉. Let g = Prol(g−, l). Then it is easy to see that g0 = 〈y, xizj , zizj \| 1 ≤ i, j ≤ n〉. 56 B. Doubrov, Y. Machida and T. Morimoto Further a simple computation shows that g = I(y, z1, . . . , zn), the ideal generated by y and z1, . . . , zn. In particular, the subspace g′ consisting of all polyno- mials in variables xi, 1 ≤ i ≤ n, forms a complementary linear subspace to g in l. Moreover, it is invariant with the action of g−. Since {xi, zj} = δij for all 1 ≤ i, j ≤ n, we see that the cohomology complex C(g−, g ′) coincides with the canonical Koszul complex, which is known to be acyclic. In particular, we get that H1(g−, g ′) = 0. • Case 2: g− is 2-graded and g−1 is non-degenerate. Let dim g−1 = 2m. Then up to Aut0(m), we can assume that g− = 〈1, xi, zi \| 1 ≤ i ≤ m〉. Computing g0 we get g0 = 〈y, xixj , xizj , zizj , xaxb, xazb, zazb〉, where 1 ≤ i, k ≤ m; m+ 1 ≤ a, b ≤ n. Further computations show that a polynomial f belongs to g if and only if it has no terms linear in (xa, za), m + 1 ≤ a ≤ n. In particular, as complementary linear subspace g′, we can choose g′ = 〈 xaPa(xi, y, zi), zaQa(xi, y, zi) 〉 , 1 ≤ i, k ≤ m; m + 1 ≤ a, b ≤ n, where Pa, Qa, m + 1 ≤ a ≤ n, are arbitrary polynomials in (xi, y, zi), 1 ≤ i ≤ m. Note that this space is invariant with respect to g− and splits into the direct sum of 2(n−m) copies of submodules, which are parametrized by a single polynomial P (xi, y, zi) each. In its turn, each such submodule is isomorphic to the graded Lie algebra l′ associated to the filtered Lie algebra of all contact vector fields on the (2m+ 1)-dimensional contact manifold. Since such Lie algebra coincides with the Tanaka prolongation of g−, we get that H1 +(g−, l ′) = 0 and hence H1 +(g−, g ′) = 0. Thus, corollary to Theorem 7.9 can be applied in both cases. Example 7.11. Consider now the extrinsic geometry of scalar 2nd order ODEs (ordinary dif- ferential equations) under contact and point transformations. We can consider a scalar 2nd order ODE as an embedding ϕ : (M, fM ) → (N, fN ), where (N, fN ) is a 4-dimensional jet space J2(R,R) with the canonical jet space filtration fN : f−1 N ⊂ f −2 N ⊂ f−3 N = TN, and M is a 3-dimensional manifold (a 2nd order ODE) such that ϕ(M) is transversal to the projection π : J2(R,R)→ J1(R,R) at each point x ∈ ϕ(M). The filtration fM is induced by the filtration fN : fiM = fiN ∩ TM for all i ∈ Z. Let m be the symbol algebra of (N, fN ). It is well-known that m = m−1 ⊕ m−2 ⊕ m−3 such that m−1 = 〈E2, X〉, m−2 = 〈E1〉, m−3 = 〈E0〉, where [X,Ei] = Ei−1, [Ei, Ej ] = 0. Note that the fibres of the projection π are tangent to the characteristic direction of f−2 N , which lies inside f−1 N and corresponds to the subspace 〈E2〉 in m−1. Extrinsic Geometry and Linear Differential Equations 57 Since M is transversal to this direction, the space ϕ∗(TxM) is a 3-dimensional vector space complementary to it, and grϕ∗(TxM) = gr(fM )ϕ(x) is a 3-dimensional graded subalgebra of m complementary to 〈E2〉. It is easy to see that all such subspaces are equivalent to the fixed subspace g− = 〈X,E1, E0〉 under the action of Aut0(m). Consider now the extrinsic invariants of the embedding ϕ under contact transformations. Let l = lcont be is the (infinite-dimensional) graded Lie algebra associated with the filtered Lie algebra of all vector fields on N preserving the filtration fN . Due to Lie’s theorem all such vector fields are the lifts of the contact vector fields on J1(R,R). As in the previous example, the Lie algebra l can be described as a set of all contact vector fields Xf , where f runs through all polynomials in (x, y, z). The grading of l is defined as follows. First, define weighted degree on R[x, y, z] by setting deg x = 1, deg y = 3, deg z = 2. Then for any weighted homogeneous polynomial f ∈ R[x, y, z] we define degXf = deg f − 3. In the following we identify l with the space of such polynomials equipped with the above Lie bracket. Remark 7.12. This construction easily generalizes to the description of the graded Lie alge- bra associated with the infinite-dimensional Lie algebra of vector fields preserving the contact distribution on Jk(R,R) for any k ≥ 1. Namely, in the case the (abstract) Lie algebra is still isomorphic to l, but the grading is defined differently according to the rules: deg x = 1, deg y = k + 1, deg z = k, and degXf = deg f − k − 1. In this notation, the Lie algebra m = l− has the form 〈1, x, x2, z〉. The subalgebra g− then corresponds to the subspace g− = 〈1, x, z〉. Let us compute g = Prol(g−, l). It is easy to see that l0 = 〈 y, x3, xz 〉 , g0 = 〈y, xz〉, l1 = 〈 xy, x4, x2z, z2 〉 , g1 = 〈 x(y − xz), z2 〉 . Further simple computations show that g = {P (z, y − xz) +Q(z, y − xz)x}, where P , Q are two arbitrary polynomials in 2 variables. Note that g coincides with all contact polynomial symmetries of the trivial differential equation y′′ = 0, or, in other words, with all polynomial vector fields on J1(R,R) which preserve the 1-dimensional vector distribution 〈d/dx〉. Let us show that H1 +(g−, l/g) = 0. The short exact sequence 0→ g→ l→ l/g→ 0 induces the long exact sequence · · · → H1(g−, l)→ H1(g−, l/g)→ H2(g−, g)→ · · · . (7.1) So, it is sufficient to prove that H1 +(g−, l) = 0 and H2 +(g−, g) = 0. This is done in two steps below. 58 B. Doubrov, Y. Machida and T. Morimoto Step 1. Denote by g−[−1] ⊂ l[−1] the same abstract Lie algebras, but with the gra- ding induced from J1(R,R), that is by assuming that deg x = 1, deg y = 2, deg z = 1 and degXf = deg f − 2. Then it is well known that l[−1] is the Tanaka prolongation of g−[−1] and hence H1 r (g−[−1], l[−1]) = 0 for all r ≥ 0. We note that H1 (g−[−1], l[−1]) may have non-zero components in negative degree. But it is easy to check that the negative part of C1 (g−[−1], l[−1]) = Hom (g−[−1], l[−1]) belongs to the negative part of C1(g−, l). Hence, we also get H1 +(g−, l) = 0. Step 2. Let us prove that H2 +(g−, g) = 0. The Legendre transform (x, y, z) 7→ (−z, y − xz, x) induces an automorphism of l that maps this vector distribution to 〈∂/∂z〉 and, hence, maps the subalgebra g to the subalgebra of all polynomial vector fields on the plane (x, y) lifted to J1(R,R). Thus, computingH(g−, g) we can identify g with the Lie algebra of all polynomial vector fields on the plane with the grading induced from the conditions deg x = 2, deg y = 3. In particular, under this identification we have: g−3 = 〈∂/∂y〉, g−2 = 〈∂/∂x〉, g−1 = 〈x∂/∂y〉. We note that V = 〈∂/∂x, ∂/∂y〉 forms an ideal in g−. Moreover, the cohomology complex for the V -module g is exactly the standard Koszul cohomology complex, which is known to be acyclic. Thus, we have Hq(V, g) = 0 for all q > 0 and H0(V, g) = V . Now use Serre–Hochschild spectral sequence [10] associated with the ideal V ⊂ g−. Its second term Ep,q2 is equal to Ep,q2 = Hp ( g−/V,H q(V, g) ) . Since we want to compute H2(g−, g), we can assume that p+ q = 2. If q > 0, then by the above Hq(V, g) = 0. Finally, if q = 0 and p = 2, then H2 ( g−/V,H 0(V, g) ) = 0, since g−/V is 1-dimensional. This proves that H2(g−, g) = 0. In particular, this implies that an arbitrary 2nd order ODE is contactly equivalent to the trivial equation y′′ = 0. Consider now the case of the pseudogroup of point transformations on the plane. Let l = lpoint be an infinite-dimensional graded Lie algebra associated with the (filtered) Lie algebra of all infinitesimal point transformations acting on J2(R,R). It is well-known that lpoint can be identified with the subalgebra of lcont consisting of all vector fields Xf , where f is a polynomial in x, y, z linear in z. We note that the grading of l is induced from its prolongation to J2(R,R) and the standard filtration on J2(R,R) (see above), which is different from the more conventional grading on the Lie algebra of polynomial vector fields on the place. We have the same g− ⊂ l− as in the contact case. But the prolongation is now different. Similar to the contact case, it is easy to show by direct computation that the prolongation of g− in lpoint coincides with the point symmetry algebra of the trivial equation y′′ = 0, which is exactly the Lie algebra sl(3,R) of all projective vector fields on the plane: g−3 = 〈∂/∂y〉, g−2 = 〈∂/∂x〉, g−1 = 〈x∂/∂y〉, g0 = 〈x∂/∂x, y∂/∂y〉, g1 = 〈y∂/∂x〉, g2 = 〈xU〉, g3 = 〈yU〉, where U = x∂/∂x+ y∂/∂y. Extrinsic Geometry and Linear Differential Equations 59 As above, we can compute H1 +(g−, l/g) using the long exact sequence (7.1). Next, we can prove that Hk +(g−, l) = 0 for k = 1 and k = 2 by means of the Serre–Hochschild spectral sequence as in Step 2 above. 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Soc. 8 (1907), 233–260. https://doi.org/10.2206/kyushumfs.47.283 https://doi.org/10.4099/math1924.9.293 https://doi.org/10.14492/hokmj/1381517794 https://doi.org/10.14492/hokmj/1381413178 https://doi.org/10.2969/aspm/03710205 https://doi.org/10.1017/CBO9780511543142 https://doi.org/10.1017/CBO9780511543142 https://doi.org/10.1016/j.difgeo.2008.04.017 https://arxiv.org/abs/math.DG/0608471 https://doi.org/10.2969/aspm/02510318 https://doi.org/10.2969/aspm/02510318 https://doi.org/10.2748/tmj/1178227829 https://doi.org/10.14492/hokmj/1381517803 https://doi.org/10.2969/aspm/02210265 https://doi.org/10.1007/BF02787690 https://doi.org/10.1215/kjm/1250523814 https://doi.org/10.14492/hokmj/1381758416 https://doi.org/10.2307/1986347 1 Introduction 2 Filtered manifolds and flag varieties 2.1 Filtered manifolds 2.2 Flag varieties 2.3 Osculating maps 3 Three categories 3.1 Category of L/L0 extrinsic geometries 3.2 Category of L/L0 differential equations 3.3 Category of L/L0 extrinsic bundles 3.4 Congruence classes 4 Categorical isomorphisms 4.1 From an extrinsic geometry to an extrinsic bundle 4.2 From an extrinsic bundle to an extrinsic geometry 4.3 From a differential equation to an extrinsic bundle 4.4 From an extrinsic bundle to a differential equation 4.5 From an extrinsic geometry to a differential equation 4.6 From a differential equation to an extrinsic geometry 4.7 L/L0 differential equations as linear differential equations in weighted jet bundles 4.8 Dual embeddings and differential equations 4.9 Examples 5 Equivalence problems, extrinsic normal Cartan connections and invariants 5.1 Relative prolongations, standard models and extrinsic cohomology groups 5.2 Auxiliary groups, complementary subspaces 5.3 First reduction, bundle Q(0) 5.4 Structure function chi 5.5 Condition (C) 5.6 Reductions 6 Rigidity of rational homogeneous varieties 6.1 Extrinsic parabolic geometries 6.2 Harmonic theory on semisimple Lie algebras 6.3 Normal reductions 6.4 Vanishing of cohomology groups and rigidity 6.5 Simple Lie algebras of rank 2 and their adjoint representations 6.6 Realizability of deformations 7 Equivalence problem of extrinsic geometries – general case 7.1 Generalizations to the non-principal and to the infinite-dimensional cases 7.2 Examples References
id	nasplib_isofts_kiev_ua-123456789-211362
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn	1815-0659
language	English
last_indexed	2026-03-15T03:02:41Z
publishDate	2021
publisher	Інститут математики НАН України
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spelling	Doubrov, Boris Machida, Yoshinori Morimoto, Tohru 2025-12-30T15:56:04Z 2021 Extrinsic Geometry and Linear Differential Equations. Boris Doubrov, Yoshinori Machida and Tohru Morimoto. SIGMA 17 (2021), 061, 60 pages 1815-0659 2020 Mathematics Subject Classification: 53A55; 53C24; 53C30; 53D10 arXiv:1904.05687 https://nasplib.isofts.kiev.ua/handle/123456789/211362 https://doi.org/10.3842/SIGMA.2021.061 We give a unified method for the general equivalence problem of extrinsic geometry, based on our formulation of a general extrinsic geometry as that of an osculating map : ( , f) → / ⁰ ⊂ Flag( , ) from a filtered manifold ( , f) to a homogeneous space / ⁰ in a flag variety Flag( , ), where L is a finite-dimensional Lie group and ⁰ its closed subgroup. We establish an algorithm to obtain complete systems of invariants for the osculating maps that satisfy a reasonable regularity condition, namely, a constant symbol of type ( ₋, gr , ). We show the categorical isomorphism between the extrinsic geometries in flag varieties and the (weighted) involutive systems of linear differential equations of finite type. Therefore, we also obtain a complete system of invariants for a general involutive system of linear differential equations of finite type and of constant symbol. The invariants of an osculating map (or an involutive system of linear differential equations) are proved to be controlled by the cohomology group ¹₊( ₋, / ¯) which is defined algebraically from the symbol of the osculating map (resp. involutive system), and which, in many cases (in particular, if the symbol is associated with a simple Lie algebra and its irreducible representation), can be computed by the algebraic harmonic theory, and the vanishing of which gives rigidity theorems in various concrete geometries. We also extend the theory to the case when is infinite-dimensional. The third author is partially supported by JSPS KAKENHI Grant Number 17K05232. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Extrinsic Geometry and Linear Differential Equations Article published earlier
spellingShingle	Extrinsic Geometry and Linear Differential Equations Doubrov, Boris Machida, Yoshinori Morimoto, Tohru
title	Extrinsic Geometry and Linear Differential Equations
title_full	Extrinsic Geometry and Linear Differential Equations
title_fullStr	Extrinsic Geometry and Linear Differential Equations
title_full_unstemmed	Extrinsic Geometry and Linear Differential Equations
title_short	Extrinsic Geometry and Linear Differential Equations
title_sort	extrinsic geometry and linear differential equations
url	https://nasplib.isofts.kiev.ua/handle/123456789/211362
work_keys_str_mv	AT doubrovboris extrinsicgeometryandlineardifferentialequations AT machidayoshinori extrinsicgeometryandlineardifferentialequations AT morimototohru extrinsicgeometryandlineardifferentialequations

Extrinsic Geometry and Linear Differential Equations

Institution

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