Monge-Ampère Systems with Lagrangian Pairs

The classes of Monge-Ampère systems, decomposable and bi-decomposable Monge-Ampère systems, including equations for improper affine spheres and hypersurfaces of constant Gauss-Kronecker curvature are introduced. They are studied by the clear geometric setting of Lagrangian contact structures, based...

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Veröffentlicht in:	Symmetry, Integrability and Geometry: Methods and Applications
Datum:	2015
Hauptverfasser:	Ishikawa, G., Machida, Y.
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Sprache:	Englisch
Veröffentlicht:	Інститут математики НАН України 2015
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Zitieren:	Monge-Ampère Systems with Lagrangian Pairs / G. Ishikawa, Y. Machida // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 31 назв. — англ.

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author	Ishikawa, G. Machida, Y.
author_facet	Ishikawa, G. Machida, Y.
citation_txt	Monge-Ampère Systems with Lagrangian Pairs / G. Ishikawa, Y. Machida // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 31 назв. — англ.
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container_title	Symmetry, Integrability and Geometry: Methods and Applications
description	The classes of Monge-Ampère systems, decomposable and bi-decomposable Monge-Ampère systems, including equations for improper affine spheres and hypersurfaces of constant Gauss-Kronecker curvature are introduced. They are studied by the clear geometric setting of Lagrangian contact structures, based on the existence of Lagrangian pairs in contact structures. We show that the Lagrangian pair is uniquely determined by such a bi-decomposable system up to the order, if the number of independent variables ≥3. We remark that, in the case of three variables, each bi-decomposable system is generated by a non-degenerate three-form in the sense of Hitchin. It is shown that several classes of homogeneous Monge-Ampère systems with Lagrangian pairs arise naturally in various geometries. Moreover we establish the upper bounds on the symmetry dimensions of decomposable and bi-decomposable Monge-Ampère systems respectively in terms of the geometric structure and we show that these estimates are sharp (Proposition 4.2 and Theorem 5.3).
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 11 (2015), 081, 32 pages Monge–Ampère Systems with Lagrangian Pairs Goo ISHIKAWA † and Yoshinori MACHIDA ‡ † Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan E-mail: ishikawa@math.sci.hokudai.ac.jp ‡ Numazu College of Technology, 3600 Ooka, Numazu-shi, Shizuoka, 410-8501, Japan E-mail: machida@numazu-ct.ac.jp Received April 10, 2015, in final form October 05, 2015; Published online October 10, 2015 http://dx.doi.org/10.3842/SIGMA.2015.081 Abstract. The classes of Monge–Ampère systems, decomposable and bi-decomposable Monge–Ampère systems, including equations for improper affine spheres and hypersurfaces of constant Gauss–Kronecker curvature are introduced. They are studied by the clear geo- metric setting of Lagrangian contact structures, based on the existence of Lagrangian pairs in contact structures. We show that the Lagrangian pair is uniquely determined by such a bi-decomposable system up to the order, if the number of independent variables ≥ 3. We remark that, in the case of three variables, each bi-decomposable system is generated by a non-degenerate three-form in the sense of Hitchin. It is shown that several classes of homo- geneous Monge–Ampère systems with Lagrangian pairs arise naturally in various geometries. Moreover we establish the upper bounds on the symmetry dimensions of decomposable and bi-decomposable Monge–Ampère systems respectively in terms of the geometric structure and we show that these estimates are sharp (Proposition 4.2 and Theorem 5.3). Key words: Hessian Monge–Ampère equation; non-degenerate three form; bi-Legendrian fibration; Lagrangian contact structure; geometric structure; simple graded Lie algebra 2010 Mathematics Subject Classification: 58K20; 53A15; 53C42 1 Introduction 1.1. The second-order partial differential equation Hess(f) = det ( ∂2f ∂xi∂xj ) 1≤i,j≤n = c (c is constant, c 6= 0), for a scalar function f of n real variables xi, i = 1, 2, . . . , n, describes improper (parabolic) affine hyperspheres z = f(x1, . . . , xn) and it plays a significant role in equi-affine geometry (see [25] for example). Similarly the equation of constant Gaussian (Gauss–Kronecker) curvature K = c (c is constant) for hypersurfaces is important in Riemannian geometry (see [17] for example). Note that it is written, for graphs z = f(x1, . . . , xn), as the equation Hess(f) = (−1)nc ( 1 + p2 1 + · · ·+ p2 n )n+2 2 , where pi = ∂f ∂xi . Therefore the equations Hess(f) = c and K = c are regarded as Monge–Ampère equations, and they are studied from geometric aspects in this paper. If we treat these equations in the framework of Monge–Ampère systems, then we realize that they have a specific character. In [12], we treated improper affine spheres and constant Gaussian curvature surfaces in R3 from the view point of Monge–Ampère equations of two variables, and we analyzed the singula- rities of their geometric solutions. There we effectively used the direct sum decomposition of the mailto:ishikawa@math.sci.hokudai.ac.jp mailto:machida@numazu-ct.ac.jp http://dx.doi.org/10.3842/SIGMA.2015.081 2 G. Ishikawa and Y. Machida standard contact structure D ⊂ TR5 on R5 into a pair of two Lagrangian plane fields E1, E2, namely a Lagrangian pair. Based on the notion of Lagrangian pairs generalized to the higher-dimensional cases, namely, for contact manifolds of dimension 2n + 1, we introduce decomposable and bi-decomposable Monge–Ampère systems with Lagrangian pairs in Section 2. A decomposable (resp. a bi- decomposable) Monge–Ampère system is defined by a decomposable n-form (resp. a sum of two decomposable n-forms) which is compatible with the underlying Lagrangian pair. The class of Monge–Ampère systems with Lagrangian pairs, which is introduced in this paper, is invariant under contact transformations. If a Monge–Ampère system is isomorphic to a Monge– Ampère system with a Lagrangian pair, then it is accompanied with a Lagrangian pair (see Definition 2.6). On the other hand, it is not trivial that the Lagrangian pair is uniquely associa- ted to a given Monge–Ampère system. Then we are led to natural questions: Is the Lagrangian pair uniquely determined by the decomposable (resp. the bi-decomposable form) form? Is the Lagrangian pair recovered only from the Monge–Ampère system? We see that Lagrangian pair is not determined by the decomposable form. Any decomposable Monge–Ampère system with Lagrangian pair (E1, E2) is of Lagrangian type in the sense of [22], and the Lagrangian subbundle E1 is obtained as the characteristic system of the Monge–Ampère system (see also [24] and [7, Chapter V]). However the complementary Lagrangian subbundle E2 is not uniquely determined. In Section 3, we show a close relation between Lagrangian pairs and bi-decomposable forms, and give an answer to the above questions by showing that a bi-decomposable Monge–Ampère system has the unique n-form as a local generator up to a multiplication by a non-zero function and modulo the contact form (Theorem 3.2) and that such a bi-decomposable form uniquely determines the associated Lagrangian pair (E1, E2) uniquely, provided n ≥ 3 (Theorem 3.8). Thus we see that any automorphism of a given Monge–Ampère system with Lagrangian pair induces an automorphism of the underlying Lagrangian pair, if n ≥ 3. Note that Lagrangian pair is not determined by the bi-decomposable form if n = 2 (Remark 3.11). 1.2. It follows that the study of Monge–Ampère systems with Lagrangian pairs has close relation with the theory of Takeuchi [28] on “Lagrangian contact structures”. From the viewpoint of geometric structures, the comparison of the Lagrangian contact struc- tures and Monge–Ampère systems with Lagrangian pairs goes as follows: we treat Monge– Ampère systems with Lagrangian pairs on M = P (T ∗W ), the projective cotangent bundle over a manifold W of dimension n+ 1 in the following two cases. For the first case, if the base space W has an affine structure, then M = P (T ∗W ) has the natural Lagrangian contact structure, i.e., a Lagrangian pair, see [28]. Moreover a Monge– Ampère system with the Lagrangian pair on M is naturally induced, if W has an equi-affine structure. Here an equi-affine structure on W means that W is equipped with a torsion-free linear connection and a parallel volume form on W , see [25]. Furthermore if W is the affine flat Rn+1 or the torus Tn+1, then we have the generalization of the Hessian constant equation Hess(f) = c. For the second case, we take a Lagrangian contact structure on M = P (T ∗W ) or on the unit tangent bundle T1W over W with the projective structure induced from a Riemannian metric on W . Recall that the projective structure is defined as the equivalence class of the Levi- Civita connection, under the projective equivalence on torsion-free linear connections which is determined by the set of un-parametrized geodesics. Moreover we consider a Monge–Ampère system with Lagrangian pair on M induced from the volume of the Riemannian metric on W . Furthermore if W is a projectively flat Riemannian manifold, that is, one of the spaces En+1, Sn+1, Hn+1 with constant curvature, we obtain the generalization of the Gaussian curvature constant equation K = c as an “Euler–Lagrange” Monge–Ampère system (see Section 6 and Sections 8.2–8.5). Monge–Ampère Systems with Lagrangian Pairs 3 We summarize those subjects as the chart: Wn+1 M2n+1 = P (T ∗W )[ an equi-affine structure, the volume structure of a Riemannian metric ] ←→ a M-A system with Lagrangian pair ↓ ↓[ an affine structure, a projective structure ] ←→ a Lagrangian contact structure Here the lower row indicates the underlying structures and the upper row indicates the additional structures. 1.3. In Section 4 we recall the theory of Takeuchi. In Section 5, we study the symmet- ries of a Monge–Ampère system with a Lagrangian pair. Using the results in this paper, we show that the local automorphisms of a Monge–Ampère system with a Lagrangian pair form a finite-dimensional Lie pseudo-group, provided n ≥ 3. We determine the maximal dimension of the automorphism pseudo-groups of the Monge–Ampère systems with flat Lagrangian pairs (Theorem 5.3). Based on those aspects, we characterize a class of Monge–Ampère systems which includes the equations Hess(f) = c and K = c, c 6= 0. In fact, the class of Monge–Ampère equations of type Hess(f) = F (x1, . . . , xn, f(x), p1, . . . , pn), F 6= 0, is characterized and called the class of Hesse Monge–Ampère systems in Section 6. We observe that the class of Hesse Monge–Ampère systems is invariant under the contact transformations in the cases n ≥ 3 (Proposition 6.8). For instance, the Legendre dual of the above equation is well defined and given by Hess(f) = 1 F ( p1, . . . , pn, n∑ i=1 xipi − f(x), x1, . . . , xn ) . Note that in the case n = 2, Hess(f) = ±1 is transformed to the Laplace equation fx1x1+fx2x2=0 or to the wave equation fx1x1 − fx2x2 = 0. Therefore the class of Hesse Monge–Ampère systems is not invariant under the contact transformations in the case n = 2. In the case n = 3, any Monge–Ampère system of the class is given by a non-degenerate three- form which is decomposed uniquely up to ordering by two decomposable forms (see [3, 11, 19, 21] and also Section 3). This fact and its generalizations are the basic reasons behind the above observation. Further, we provide the unified picture of various subclasses of Monge–Ampère equations with significant examples in arbitrary dimensions from various geometric frameworks. In Section 7, we introduce the general method to construct Euler–Lagrange Monge–Ampère system. We apply the method of construction to several situations and obtain several illustrative examples. In Section 8, based on the general method, we show that homogeneous Monge–Ampère systems with flat Lagrangian pairs arise in a very natural manner, in equi-affine geometry, in Euclidean geometry, in sphere geometry, in hyperbolic geometry, and moreover in Minkowski geometry. For these geometries, we construct Monge–Ampère systems with Lagrangian pairs explicitly and globally. Moreover we show that the estimate proved in Section 5 is best possible by providing the example with the maximal symmetry. 1.4. In this paper we treat, as underlying manifolds for Monge–Ampère systems, contact manifolds of dimensions ≥ 5. We remark that 3-dimensional contact manifolds with Lagrangian 4 G. Ishikawa and Y. Machida pairs are related to second-order ordinary differential equations with normal form. They are studied in detail in [2, 14]. As in [3, 4, 19, 21], Lagrangian pairs can be formulated, at least locally, on symplectic manifolds by means of reduction process. In this paper we adopt the contact framework of Monge–Ampère equations based on Lagrangian contact structures. We remark that the paper [22] treats in detail the class of decomposable Monge–Ampère systems or Monge–Ampère systems with one decomposability, which are modeled on the Monge– Ampère equation Hess(z) = 0. We will solve the equivalence problem of Monge–Ampère systems with Lagrangian pairs in the subsequent paper. 2 Monge–Ampère systems and Lagrangian pairs We start with the general definition of Monge–Ampère systems [5, 20, 23, 24]. Recall that a contact structure D on a manifold M is a subbundle of TM of codimension one locally defined by a contact 1-form θ by D = {θ = 0} such that dθ is non-degenerate, that is, symplectic, on the bundle D. A manifold endowed with a contact structure is called a contact manifold. It is known that the dimension of a contact manifold is odd. Let (M,D) be a contact manifold of dimension 2n+ 1 with the contact structure D ⊂ TM . A Monge–Ampère system on M is by definition an exterior differential system M⊂ ΩM gene- rated locally by a contact form θ for D and an n-form ω on M : for each point x ∈ M , there exists an open neighborhood U of x in M such that, algebraically, M\|U = 〈θ, dθ, ω〉ΩU . Here ΩM (resp. ΩU ) is the sheaf of germs of exterior differential forms on M (resp. on U). In this case we call ω a local generator of the Monge–Ampère systemM (modulo the contact ideal 〈θ, dθ〉). Note that one may assume that ω is effective, i.e., dθ ∧ ω ≡ 0 (mod θ). Also note that the (n+ 1)-form dω belongs to the contact ideal locally necessarily (see [3, 4, 20]). Let D ⊂ TM be a vector bundle of rank 2n in the tangent bundle of a manifold M . Recall that a conformal symplectic structure on D is a reduction of the structure group of D to the conformal symplectic group CSp(R2n). If (M,D) is a contact manifold of dimension 2n + 1, then the conformal symplectic structure on D is defined locally by dθ for a contact form θ which gives D locally. In particular, for each point x ∈ M , Dx has the symplectic structure which is determined uniquely up to a multiplication of a non-zero constant. We call a linear subspace W ⊂ Dx Lagrangian if W is isotropic for the conformal symplectic structure on Dx and dimRW = n. A subbundle E ⊂ D is called a Lagrangian subbundle if Ex ⊂ Dx is Lagrangian for any x ∈M . Now we define the key notion in this paper. Definition 2.1. Let (M,D) be a contact manifold. A Lagrangian pair is a pair (E1, E2) of Lagrangian subbundles of D with respect to the conformal symplectic structure on D which satisfies the condition D = E1 ⊕ E2. Remark 2.2. In [6, § 5.2], the notion of bi-Lagrangian structure is defined as the transverse pair of Lagrangian foliations in a symplectic manifold. Since we treat the contact case, it might be natural to use the terminology “Legendrian” instead of “Lagrangian”. However we would like to use “Lagrangian” in the general cases, and to use the terminology “Legendrian” just for the integrable cases, such as “Legendrian submanifolds” and “Legendrian fibrations” (see Section 6). Monge–Ampère Systems with Lagrangian Pairs 5 The standard example of Lagrangian pair is given as follows. The standard example. The standard example of Lagrangian pair is given on the stan- dard Darboux model. Consider R2n+1 with coordinates (x1, . . . , xn, z, p1, . . . , pn) and with the standard contact structure Dst = {v ∈ TM \| θst(v) = 0} defined by the standard contact form θst = dz − n∑ i=1 pidxi. Then we set Est 1 = {v ∈ Dst \| dp1(v) = · · · = dpn(v) = 0} = 〈 ∂ ∂x1 + p1 ∂ ∂z , . . . , ∂ ∂xn + pn ∂ ∂z 〉 , Est 2 = {u ∈ Dst \| dx1(u) = · · · = dxn(u) = 0} = 〈 ∂ ∂p1 , . . . , ∂ ∂pn 〉 . Then (Est 1 , E st 2 ) is a Lagrangian pair on (R2n+1, Dst). Note that, in [28], Takeuchi called a contact structure D endowed with a Lagrangian pair (E1, E2) a Lagrangian contact structure (D;E1, E2) and gave a detailed study on this geometric structure. Moreover we consider an exterior differential system associated to a given Lagrangian pair or Lagrangian contact structure. Definition 2.3. A Monge–Ampère system M is called a decomposable Monge–Ampère system with a Lagrangian pair (E1, E2) if in a neighborhood of each point of M , there exists a local generator ω of M satisfying the following decomposing condition: iuω = 0 (u ∈ E1), and ω\|E2 is a volume form on E2. Such a decomposable Monge–Ampère system of Lagrangian type is a decomposable Monge– Ampère system with the characteristic system E1 in the sense of [22]. Note that this class of Monge–Ampère equations have been introduced in [9]. Also note that decomposable Monge– Ampère systems are studied from another perspective in [1] by the name of Goursat equations. Definition 2.4. A Monge–Ampère systemM is called a bi-decomposable Monge–Ampère system with a Lagrangian pair (E1, E2) if, around each point of M , there exists a local generator ω ofM of the form ω = ω1 − ω2 by n-forms ω1 and ω2 satisfying the following bi-decomposing condition: iuω1 = 0 (u ∈ E2), ivω2 = 0 (v ∈ E1), ω1\|E1 is a volume form on E1, and ω2\|E2 is a volume form on E2. We call such an n-form ω a bi-decomposable form and (ω1, ω2) a bi-decomposition of ω. Given a Lagrangian pair (E1, E2) on (M,D) and n-forms ω1, ω2 satisfying the bi-decomposing condition for (E1, E2), we define a Monge–Ampère systemM with Lagrangian pair by setting ω = ω1−ω2. Remark 2.5. An immersion f : L → M of an n-dimensional manifold L to M is called a geo- metric solution of a Monge–Ampère systemM = 〈θ, dθ, ω〉 if f∗M = 0, namely, if f∗θ = 0, i.e., f is a Legendrian immersion, and f∗ω = 0. Any geometric solution to a decomposable (resp. bi-decomposable) Monge–Ampère systemM with a Lagrangian pair D = E1 ⊕ E2 on a contact manifold (M2n+1, D) has a crucial property. In fact, an immersion f : L → M is a geometric solution of a decomposable Monge–Ampère system with a Lagrangian pair D = E1 ⊕ E2 if and only if f∗(TpL) ∩ (E1)p 6= {0}. 6 G. Ishikawa and Y. Machida For a bi-decomposable Monge–Ampère system M with a Lagrangian pair D = E1 ⊕ E2, suppose that f : L→M is a geometric solution of M. Then, for any p ∈ L, we see that E1 ∩ f∗(TpL) = {0}, if and only if E2 ∩ f∗(TpL) = {0}, equivalently, E1 ∩ f∗(TpL) 6= {0}, if and only if E2 ∩ f∗(TpL) 6= {0}. In fact, let W be an n-plane in Dp = (E1)p ⊕ (E2)p satisfying ω\|W = 0. The direct sum decomposition defines projections π1 : Dp → (E1)p and π2 : Dp → (E2)p. Then E1 ∩W = {0} if and only if π2\|W : W → (E2)p is an isomorphism, and E2 ∩W = {0} if and only if π1\|W : W → (E1)p is an isomorphism, respectively. For any bi-decomposition ω = ω1 − ω2 and for any basis u1, . . . , un of W , we have ω1(π1(u1), . . . , π1(un)) = ω1(u1, . . . , un) = ω2(u1, . . . , un) = ω2(π2(u1), . . . , π2(un)). Using the bi-decomposing condition again, we see that π1\|W is an isomorphism if and only if the most left hand side is non-zero, and it is equivalent to the condition that π2\|W is an isomorphism. Now we are led to natural questions: • Is the Lagrangian pair (E1, E2) uniquely determined by a decomposable form? • Is the Lagrangian pair (E1, E2) uniquely determined by a bi-decomposable form? • Is the Lagrangian pair (E1, E2) recovered only from the Monge–Ampère system M? As is stated in Introduction, the first question is answered negatively. To answer the second and third questions, we recall the basic definitions. Definition 2.6. Let (M,D), (M ′, D′) be contact manifolds of dimension 2n + 1, and M, M′ Monge–Ampère systems on contact manifolds (M,D), (M ′, D′) respectively. A diffeomorphism Φ: M −→ M ′ is called an isomorphism of Monge–Ampère systems if (1) Φ is a contactomor- phism, namely (Φ∗)D = D′, and (2) Φ∗M′ =M. Now suppose that contact manifolds (M,D), (M ′, D′) of dimension 2n+ 1 are endowed with Lagrangian pairs (E1, E2), (E′1, E ′ 2) respectively, namely, that the decompositions D = E1 ⊕E2 and D′ = E′1 ⊕ E′2 are given. Suppose n ≥ 3. Then, from the result in Section 3 mentioned above, we have that any isomorphism Φ of a Monge–Ampère system M with the Lagrangian pair (E1, E2) and a Monge–Ampère system M′ with the Lagrangian pair (E′1, E ′ 2) necessarily preserves the Lagrangian pairs up to ordering, namely, (Φ∗)E1 = E′1, (Φ∗)E2 = E′2 or (Φ∗)E1 = E′2, (Φ∗)E2 = E′1. In the case n = 2, a Lagrangian pair is not uniquely determined from a Monge–Ampère system and moreover, the automorphism pseudo-group may be of infinite dimension. For instance, consider the equation Hess = −1 which is isomorphic to the linear wave equation fx1x1−fx2x2 = 0 and to the equation fx1x2 = 0. Then the last equation has infinite-dimensional automorphisms induced by diffeomorphisms (x1, x2) 7→ (X1(x1), X2(x2)). 3 Lagrangian pair and bi-decomposable form Let (M,D) be a contact manifold of dimension 2n + 1 with a contact structure D ⊂ TM . We have defined in Section 2 the notion of bi-decomposing conditions and bi-decomposable forms on (M,D). Then, first we show Monge–Ampère Systems with Lagrangian Pairs 7 Lemma 3.1. Let ω be an n-form on a contact manifold (M,D), D = {u ∈ TM \| θ(u) = 0} for a local contact form θ defining D, and (E1, E2) a Lagrangian pair of D. Assume that ω is a bi-decomposable form for (E1, E2), and ω = ω1−ω2 is any bi-decomposition of ω for (E1, E2). Then locally there exists a coframe θ, α1, . . . , αn, β1, . . . , βn of T ∗M such that E1 = {v ∈ D \|β1(v) = · · · = βn(v) = 0}, E2 = {u ∈ D \|α1(u) = · · · = αn(u) = 0}, and that the n-forms ω̃1 = α1 ∧ · · · ∧ αn, ω̃2 = β1 ∧ · · · ∧ βn satisfy the bi-decomposing condition for (E1, E2) with ω̃1 ≡ ω1, ω̃2 ≡ ω2, ω ≡ ω̃1 − ω̃2 up to a multiple of θ. Proof. The proof is based on the fact that the symplectic group on a finite-dimensional sym- plectic vector space acts transitively on the set of transversal pairs of Lagrangian subspaces. Let X1, . . . , Xn and P1, . . . , Pn be local frames of E1 and E2 respectively. Let R be the Reeb vector field for a local contact form θ defining D. Recall that R is defined uniquely by iRθ = 1, iRdθ = 0. Consider the dual coframe θ, α1, . . . , αn, β1, . . . , βn of T ∗M to the frame R, X1, . . . , Xn, P1, . . . , Pn of TM . Then we see, from the bi-decomposing condition, that there exist an (n− 1)-form γ and a non-vanishing function µ on M such that ω1 = µ(α1 ∧ · · · ∧αn) + θ∧ γ. By replacing α1 by 1 µα1, we may suppose µ ≡ 1. Similarly, we have ω2 ≡ β1∧· · ·∧βn mod θ. � Next we show that M has the unique local bi-decomposable generator ω modulo θ. Theorem 3.2. Let (E1, E2) be a Lagrangian pair and ω, ω′ be two bi-decomposable n-forms for the Lagrangian pair (E1, E2) on (M,D). Assume that they generate the same Monge–Ampère system M = 〈θ, dθ, ω〉 = 〈θ, dθ, ω′〉. Then there exist locally a non-vanishing function µ and an (n − 1)-form η on M such that ω′ = µω + θ ∧ η. To show Theorem 3.2, we study, for each x ∈ M , the symplectic exterior linear algebra on the symplectic vector space V = Dx with the symplectic form Θ = dθ\|Dx and with the decomposition V = V1 ⊕ V2, V1 = (E1)x, V2 = (E2)x, of (V,Θ) into Lagrangian subspaces. Let (V,Θ) be a 2n-dimensional symplectic vector space. We say that an n-form ω ∈ ∧nV ∗ is bi-decomposable if there exist a decomposition V = V1⊕V2 of V into Lagrangian subspaces V1, V2 in V and n-forms ω1, ω2 ∈ ∧nV ∗ such that ω = ω1 − ω2, iuω1 = 0 (u ∈ V2), ivω2 = 0 (v ∈ V1), ω1\|V1 6= 0, ω2\|V2 6= 0. In this case (ω1, ω2) is called a bi-decomposition of ω. Then, similarly as the proof of Lemma 3.1, we have that there exists a basis {a1, . . . , an, b1, . . . , bn} of V such that ω1 = a∗1 ∧ · · · ∧ a∗n, ω2 = b∗1 ∧ · · · ∧ b∗n, and that V1 = 〈a1, . . . , an〉, V2 = 〈b1, . . . , bn〉, where {a∗1, . . . , a∗n, b∗1, . . . , b∗n} denotes the dual basis of {a1, . . . , an, b1, . . . , bn}. Note that V2 (resp. V1) coincides with the annihilator of a∗1, . . . , a ∗ n (resp. b∗1, . . . , b ∗ n). Moreover we see that 8 G. Ishikawa and Y. Machida there exist a symplectic basis {a1, . . . , an, b1, . . . , bn} of (V,Θ) and a non-zero constants a, b such that ω = aa∗1 ∧ · · · ∧ a∗n − bb∗1 ∧ · · · ∧ b∗n. If we replace a1, b1 by (1/b)a1, bb1 respectively, so a∗1, b∗1 by ba∗1, (1/b)b∗1, and set c = ab, then we have the following: Lemma 3.3. Let ω ∈ ∧nV ∗ be a bi-decomposable n-form on a 2n-dimensional symplectic vector space (V,Θ), and ω = ω1 − ω2 a bi-decomposition of ω. Then there exist a symplectic basis {a1, . . . , an, b1, . . . , bn} of (V,Θ) and a non-zero constant c such that ω1 = ca∗1 ∧ · · · ∧ a∗n, ω2 = b∗1 ∧ · · · ∧ b∗n, ω = ca∗1 ∧ · · · ∧ a∗n − b∗1 ∧ · · · ∧ b∗n, Θ = a∗1 ∧ b∗1 + · · ·+ a∗n ∧ b∗n. A form ϕ ∈ ∧nV ∗ on a symplectic vector space (V,Θ) is called effective if the interior product iXΘ ϕ = 0 for the 2-vector iXΘ corresponding to the 2-form Θ. Note that XΘ = a1 ∧ b1 + · · ·+ an ∧ bn in terms of the basis in Lemma 3.3. That ϕ ∈ ∧nV ∗ is effective if and only if the wedge ϕ ∧Θ with the symplectic form Θ is equal to zero [4, 20]. Then we have, by Lemma 3.3: Corollary 3.4. If ω ∈ ∧nV ∗ is bi-decomposable and ω = ω1 − ω2 is any bi-decomposition, then ω, ω1 and ω2 are all effective. We will apply the following basic result to our situation. Lemma 3.5 ([20, Theorem 1.6]). Let ω, ω′ be effective k-forms on a symplectic vector space (V,Θ), (0 ≤ k ≤ n). Suppose that, for every isotropic subspace L ⊂ V on which ω\|L = 0, the form ω′ also vanishes on L, ω′\|L = 0. Then we have ω′ = µω for some µ ∈ R. Then we have the following: Lemma 3.6. Let ω, ω′ be bi-decomposable forms on (V,Θ). Suppose that ω′ is of the form λω + φ ∧Θ for a scalar λ and an (n− 2)-form φ. Then ω′ = µω for a scalar µ. Proof. By Corollary 3.4, the n-form ω is effective and so is the n-form ω′. For every Lagrangian subspace L (Θ\|L = 0) on which ω\|L = 0, the form ω′ also vanishes. Therefore, by Lemma 3.5, it follows that ω′ = µω for a scalar µ. � Remark 3.7. Lemma 3.6 can be shown, as an alternative proof, by applying Lefschetz iso- morphism Θ2 : ∧n−2 V → ∧n+2V . In fact, from ω′ − µω = φ ∧ Θ we have φ ∧ Θ2 = 0, which implies φ = 0. Proof of Theorem 3.2. Since ω′ belongs to M = 〈θ, dθ, ω〉, we set ω′ = λω + dθ ∧ φ+ θ ∧ η, for a function λ, an (n − 2)-form φ and an (n − 1)-form η on M . For each x ∈ M , we have ω′\|Dx = λ(x)ω\|Dx +Θ∧φ\|Dx , where Θ = dθ\|Dx . Then, by Lemma 3.6, we have ω′\|Dx = µ(x)ω\|Dx for a scalar µ(x) depending on x ∈M . Since ω\|(E1)x is a volume form, we see that µ(x) is unique and of class C∞. Moreover there exists a C∞ (n− 1)-form η such that ω′ − µω = θ ∧ η, which implies the required consequence. � Moreover we show the following basic result: Monge–Ampère Systems with Lagrangian Pairs 9 Theorem 3.8. Assume that n ≥ 3. Let M = 〈θ, dθ, ω〉 be a Monge–Ampère system with a Lag- rangian pair locally generated by a bi-decomposable n-form ω. Then M canonically determines the Lagrangian pair (E1, E2). Namely, if (ω1, ω2) (resp. (ω′1, ω ′ 2)) is a bi-decomposition of ω for a Lagrangian pair (E1, E2) (resp. (E′1, E ′ 2)) enjoying the bi-decomposing condition. Then we have E′1 = E1, E′2 = E2, or E′1 = E2, E′2 = E1. Theorem 3.8 follows immediately from the following: Proposition 3.9. Assume that n ≥ 3. Let (V,Θ) be a 2n-dimensional symplectic vector space. Let ω = ω1 − ω2 be a bi-decomposable n-form for a Lagrangian pair (V1, V2) of V . Then the bi- decomposition (ω1, ω2) of ω is unique up to ordering: If ω = ω′1−ω′2 is another bi-decomposition of ω for another Lagrangian pair (V ′1 , V ′ 2) of V , then V ′1 = V1, V ′2 = V2, ω′1 = ω1, ω′2 = ω2, or V ′1 = V2, V ′2 = V1, ω′1 = −ω2, ω′2 = −ω1. The bi-decomposition of ω (Proposition 3.9) is given by using the symplectic structure, based on Hitchin’s result [11, Propositions 2.1, 2.2], in the case that n is odd and n ≥ 3, as follows: Let ω = ω1 − ω2, with ω1 = ca∗1 ∧ · · · ∧ a∗n and ω2 = b∗1 ∧ · · · ∧ b∗n, as in Lemma 3.3. Let ε = a∗1 ∧ · · · ∧ a∗n ∧ b∗1 ∧ · · · b∗n be the associated basis vector for Λ2nV ∗, which is the intrinsically defined volume form by the symplectic structure on V . From the isomorphism A : Λ2n−1V ∗ −→ V ⊗ ∧2nV ∗ induced by the exterior product V ∗ ⊗ ∧2n−1V ∗ → ∧2nV ∗, we define, for each ψ ∈ ∧nV ∗, a linear transformation Kψ = Kε ψ : V −→ V by Kψ(u)ε = A(iu(ψ) ∧ ψ), and put λ(ψ) = λε(ψ) = 1 2n tr(K2 ψ). In particular, we set ψ = ω. Then we have Kωai = −cai, Kωbi = (−1)n+1cbi = cbi. Then we have K2 ω = c2 id, therefore λ(ω) = c2. Since K∗ωa ∗ i = −ca∗i , K∗ωb∗i = cb∗i , we have K∗ωω = c(−c)na∗1 ∧ · · · ∧ a∗n − cnb∗1 ∧ · · · ∧ b∗n = (−c)n(ω1 + ω2). Thus, from ω = ω1 − ω2, − 1 cnK ∗ ωω = ω1 + ω2 and λ(ω) = c2 (c > 0, < 0), we see ω1 = 1 2 ( ω ∓ λ(ω)− n 2K∗ωω ) , ω2 = 1 2 ( −ω ∓ λ(ω)− n 2K∗ωω ) . Since K∗ωω and λ(ω) are intrinsically determined from the symplectic structure on V , so is the decomposition of ω. To verify Proposition 3.9 in general case, we observe a fact from projective geometry of Plücker embeddings. Consider Grassmannian Gr(n, V ∗) consisting of all n-dimensional subspaces in the 2n-dimensional vector space V ∗, and its Plücker embedding Gr(n, V ∗) ↪→ P (∧nV ∗). Lemma 3.10. Let Λ1,Λ2 ∈ Gr(n, V ∗) with Λ1 ∩ Λ2 = {0}. (1) The projective line in P (∧nV ∗) through the two points Λ1, Λ2 does not intersect with Gr(n, V ∗) at other points. (2) Assume n ≥ 3. Let Λ3 be another point in Gr(n, V ∗) different from Λ1, Λ2. Then the projective plane in P (∧nV ∗) through the three points Λ1, Λ2, Λ3 does not intersect with Gr(n, V ∗) at other points. Proof. Choose a basis e1, . . . , en of Λ1 and en+1, . . . , e2n of Λ2 to form a basis of V ∗. Let (pi1,i2,...,in) denote Plücker coordinates of the Plücker embedding. Here i1, i2, . . . , in are n-tuple 10 G. Ishikawa and Y. Machida chosen from {1, 2, . . . , 2n}. For any sequences 1 ≤ j1 < j2 < · · · < jn−1 ≤ 2n of (n − 1)-letters and 1 ≤ k1 < k2 < · · · < kn+1 ≤ 2n of (n+ 1)-letters, we have the Plücker relation n+1∑ `=1 (−1)`pj1,j2,...,jn−1,k`pk1,k2,...,k̆`,...,kn+1 = 0 (see [8, 10] for instance). To see (1), take a point W ∈ Gr(n, V ∗) on the projective line through Λ1, Λ2. Since Plücker coordinates of Λ1 (resp. Λ2) are given by p1,...,n = 1 (resp. pn+1,...,2n = 1) with zero other coordinates, we have that Plücker coordinates of W are given by p1,...,n = λ, pn+1,...,2n = µ for some λ, µ, other coordinates pi1,...,in being null. Then, by applying the Plücker relation for (j1, . . . , jn−1) = (1, 2, . . . , n−1) and (k1, k2, . . . , kn+1) = (n−1, n, . . . , 2n), we have λµ = 0. Therefore λ = 0 or µ = 0, namely, W = Λ1 or W = Λ2. To see (2), let p̄ = (p̄i1,i2,...,in) be coordinates of Λ3. Then the coordinates p = (pi1,i2,...,in) for a point W on the plane through Λ1, Λ2, Λ3 are given by p1,...,n = λ+ p̄1,...,n, pn+1,...,2n = µ+ p̄n+1,...,2n, pi1,...,in = p̄i1,...,in , {i1, . . . , in} 6= {1, . . . , n}, {n + 1, . . . , 2n}. Suppose W ∈ Gr(n, V ∗), W 6= Λ1,Λ2,Λ3. Then we see that both p and p̄ satisfy the Plücker relations and that λµ 6= 0. Let {i1, . . . , in} 6= {1, . . . , n}, {n + 1, . . . , 2n}. Then, because n ≥ 3, there exist `, `′, ` 6= `′, such that i` 6∈ {n + 1, . . . , 2n} and i`′ 6∈ {n + 1, . . . , 2n} or that i` 6∈ {1, . . . , n} and i`′ 6∈ {1, . . . , n}. In the former case, we take (i1, . . . , ĭk, . . . , in), (ik, n+ 1, . . . , 2n), and write the Plücker relation to get µ · p̄i1,...,in = 0. In the latter case, we take (i1, . . . , ĭk, . . . , in), (1, . . . , n, ik) to get λ · p̄i1,...,in = 0. Therefore we obtain that p̄i1,...,in = 0 for {i1, . . . , in} 6= {1, . . . , n}, {n+ 1, . . . , 2n}. This means that Λ3 lies on the projective line through Λ1, Λ2, which contradicts (1). � Proof of Proposition 3.9. Set ω1−ω2 = ω′1−ω′2. Then ω1−ω2−ω′1 is equal to a decomposable form −ω′2. Let Λ1, Λ2, Λ′1, Λ′2 be points in Gr(n, V ∗) ⊂ P (∧nV ∗) corresponding to ω1, ω2, ω′1, ω′2 respectively. Then the projective plane through Λ1, Λ2, Λ′1 intersects with Gr(n, V ∗) also at Λ′2. Note that Λ′1 6= Λ′2 as well as Λ1 6= Λ2. Suppose Λ′1 6= Λ1,Λ2. By Lemma 3.10(2), we have Λ′2 = Λ1 or Λ′2 = Λ2. Then we have that Λ′1 lies on the line through Λ1, Λ2. By Lemma 3.10(1), we conclude that Λ′1 = Λ1 or Λ′1 = Λ2. If ω′1 = λω1 for some λ 6= 0, then V ′2 = V2 as the annihilator of ω1 or ω′1. If ω′1 = µω2 for some µ 6= 0, then we have V ′2 = V1. By the symmetric argument, we obtain also that V ′1 = V1 or V ′1 = V2. Thus we have V ′1 = V1, V ′2 = V2 or V ′1 = V2, V ′2 = V1. Assume V ′1 = V1, V ′2 = V2, then, restricting ω to V = V1 ⊕ V2, we have ω′1 = ω1, ω′2 = ω2. Assume V ′1 = V2, V ′2 = V1, then we have ω′1 = −ω2, ω′2 = −ω1. � Remark 3.11. If n = 2, that is, M is of dimension 5, then Theorem 3.8 does not hold. In fact, consider M = R5 with coordinates (x, y, z, p, q) and with the contact form θ = dz − pdx− qdy. Take a 2-form ω = dx ∧ dy − dp ∧ dq. Then decomposable 2-forms ω1 = dx∧dy, ω2 = dp∧dq satisfy ω = ω1−ω2 and the bi-decomposing condition for the Lagrangian pair given by E1 = { v ∈ TR5 \| θ(v) = dp(v) = dq(v) = 0 } = 〈 ∂ ∂x + p ∂ ∂z , ∂ ∂y + q ∂ ∂z 〉 , E2 = { u ∈ TR5 \| θ(u) = dx(u) = dy(u) = 0 } = 〈 ∂ ∂p , ∂ ∂q 〉 . Monge–Ampère Systems with Lagrangian Pairs 11 Then we can find other decomposable 2-forms ω′1 = d(x+ p)∧ dy, ω′2 = dp∧ d(y + q) satisfying ω = ω′1 − ω′2 and the bi-decomposing condition for another Lagrangian pair given by E′1 = { v ∈ TR5 \| θ(v) = dp(v) = d(y + q)(v) = 0 } = 〈 ∂ ∂x + p ∂ ∂z , ∂ ∂y + q ∂ ∂z − ∂ ∂q 〉 , E′2 = { u ∈ TR5 \| θ(u) = d(x+ p)(u) = dy(u) = 0 } = 〈 ∂ ∂x + p ∂ ∂z − ∂ ∂p , ∂ ∂q 〉 . Lemma 3.10(2) does not hold in the case n = 2, because Gr(2,R4) ↪→ P (∧2(R4)) = P 5 is a hypersurface and a projective plane intersects with Gr(2,R4) in infinite points (a planer curve). From the above-mentioned propositions, it follows that, in the case n ≥ 3, the notion of a bi-decomposable Monge–Ampère system with a Lagrangian pair (E1, E2) is nothing but the notion of a Monge–Ampère system generated by a bi-decomposable form ω = ω1 − ω2. 4 Lagrangian contact structures Let us recall Takeuchi’s paper [28] for Lagrangian contact structures. A contact structure with a Lagrangian pair is called a Lagrangian contact structures in [28]. A typical example of Lagrangian contact structures is the projective cotangent vector bundle M = P (T ∗W ) of an (n + 1)-dimensional manifold W with an affine structure (a torsion-free linear connection) or a projective structure (a projective equivalence class of torsion-free linear connections). For the canonical contact structure D on M , we take as a Lagrangian pair horizontal and vertical vector bundles (cf. Example 6.3). In [28], it is given the description of Cartan connections on P (T ∗W ) associated to Lagrangian contact structures and the equivalence of the vanishing of the curvature of Cartan connection and the projective flatness of W . The flat model, which is a homogeneous space qualified as a model for a Cartan connection, with Lagrangian contact structure is the projective cotangent bundle P (T ∗Pn+1) of the (n+ 1)- dimensional projective space Pn+1 = P (Rn+2). Put G = PGL(n+2,R) = GL(n+2,R)/C (C is the center, C = R× ·In+2), and g = LieG ∼= sl(n+ 2,R). The Lie algebra g has a structure of a simple graded Lie algebra (GLA) of second kind as follows: g = sl(n+ 2,R) = g−2 ⊕ g−1 ⊕ g0 ⊕ g1 ⊕ g2 =  0 0 0 0 On 0 a 0 0 ⊕   0 0 0 b1 On 0 0 tb2 0 ⊕  α 0 0 0 A 0 0 0 β  ⊕  0 tc1 0 0 On c2 0 0 0 ⊕  0 0 d 0 On 0 0 0 0  , (a, d, α, β ∈ R, b1, b2, c1, c2 ∈ Rn, A ∈ gl(n,R); α+ β + trA = 0), [gp, gq] ⊂ gp+q. Put m = g−2 ⊕ g−1, then m is a fundamental GLA of contact type, i.e., Heisenberg algebra. Put g′ = g0 ⊕ g1 ⊕ g2. 12 G. Ishikawa and Y. Machida Let G′ be the Lie subgroup of G = PGL(n+ 2,R) defined by G′ = P  ∗ ∗ ∗0 ∗ ∗ 0 0 ∗  ∈ GL(n+ 2,R)  . Then the Lie algebra of G′ is given by g′. Note that dim g′ = n2 + 2n+ 2. The group G transitively acts on the flag manifold P ( T ∗Pn+1 ) = { V1 ⊂ Vn+1 ⊂ Rn+2 \| dimV1 = 1,dimVn+1 = n+ 1 } ⊂ Pn+1 × Pn+1∗. Then G′ is the isotropy group of (V1, Vn+1) = (〈e0〉, 〈e0, . . . , en〉) for the standard basis e0, e1, . . . , en, en+1 of Rn+1. Therefore we have G/G′ ∼= P ( T ∗Pn+1 ) ( ∼= P ( T ∗Pn+1∗)). Note that g−1 ⊂ m = To(G/G ′), where o = G′ is the origin, defines the contact structure D on G/G′ which corresponds to the canonical contact structure on P (T ∗Pn+1) via the above diffeomorphism (cf. [12]). Next, we consider e1 =   0 0 0 b1 On 0 0 0 0  , e2 =  0 0 0 0 On 0 0 tb2 0  . Then we have g−1 = e1 ⊕ e2, [e1, e1] = [e2, e2] = 0, g−2 = [e1, e2], [g0, e 1] ⊂ e1, [g0, e 2] ⊂ e2. Denote by Eij ∈ gl(n+ 2,R) the matrix unit of (i, j) component. Then we put γ = En+2,1 ∈ g−2, ei = Ei+1,1 ∈ e1 ⊂ g−1, fi = En+2,i+1 ∈ e2 ⊂ g−1 (1 ≤ i ≤ n). We set [X,Y ] = −A(X,Y )γ (X,Y ∈ g−1). It follows that A(ei, fj) = δij , A(ei, ej) = 0, A(fi, fj) = 0 (1 ≤ i, j ≤ n). Therefore we have that A is a symplectic form. Then g−1 becomes a symplectic vector space with respect to A, and e1, . . . , en, f1, . . . , fn form a symplectic basis of the symplectic vector space (g−1, A). Moreover e1, e2 form a Lagrangian pair of (g−1, A). Moreover, put a1 = e1 + g′, a2 = e2 + g′. Then we easily verify that Ad(G′)a1 = a1, Ad(G′)a2 = a2, a1 ∩ a2 = g′. Monge–Ampère Systems with Lagrangian Pairs 13 Thus a1 and a2 induce invariant differential systems E1 and E2 on G/G′ respectively. The pair (E1, E2) forms a Lagrangian pair of (G/G′, D) and thus that of the standard contact structure D on P (T ∗Pn+1). Moreover both E1 and E2 are completely integrable. It follows that P (T ∗Pn+1)/E2 ∼= Pn+1, P (T ∗Pn+1)/E1 ∼= Pn+1∗, where E1, E2 are the foliations induced by E1, E2 respectively. For the linear isotropy representation ρ : G′ −→ GL(m) at the origin o = G′ in G/G′, we have, with respect to the basis {γ, e1, . . . , en, f1, . . . , fn} in m, G̃ = ρ(G′) =   a 0 0 b1 A On b2 On a tA−1 ∣∣∣∣∣ a ∈ R∗, A ∈ GL(n,R), b1, b2 ∈ Rn  . Then the G̃-structures of type m are in bijective correspondence with the Lagrangian contact structures [28, Theorem 5.1]. Note that G̃-structure is of infinite type [16, Chapter I]. However g is the prolongation of (m, g0) in the sense of Tanaka [30, 31]. Moreover G̃ = G# 0 in the notation of [29]. Thus, by the finiteness theorem of Tanaka [29, Corollary 2], we have: Proposition 4.1. Let (M,D) be a contact manifold of dimension 2n + 1 with a Lagrangian pair (E1, E2). Then the automorphism pseudo-group of all compatible diffeomorphisms on M as a Lagrangian contact structure is of finite type, that is to say, it is a finite-dimensional Lie pseudo-group. The maximum dimension of the automorphism pseudo-groups, fixing n, is given by dim sl(n+ 2,R) = (n+ 2)2 − 1 which the flat model attains. Moreover, using a result in [22], we have: Proposition 4.2. The equivalence class of a decomposable Monge–Ampère system with a Lag- rangian pair (E1, E2) on a contact manifold (M,D) is uniquely determined by the Lagrangian contact structure (D,E1, E2). Therefore the maximal dimension of the automorphism pseudo- groups of decomposable Monge–Ampère systems on (M2n+1, D) is equal to (n + 2)2 − 1. The maximum is attained by the decomposable Monge–Ampère system Hess = 0 on the flat model on M = PT ∗(Pn+1). In fact it is given the following result in [22], which implies Proposition 4.2: Lemma 4.3 ([22, Proposition 2.1]). Let (V,Θ) be a symplectic vector space of dimension 2n. For given two non-zero decomposable n-covectors ω = β1 ∧ · · · ∧ βn, ω′ = β′1 ∧ · · · ∧ β′n with βi, β ′ i ∈ V ∗, we have ω′ = λω+φ∧Θ for a nonzero scalar λ and a (n−2)-covector φ, if and only if, the annihilator of β1, . . . , βn in V and the annihilator of β′1, . . . , β ′ n in V are either identical or perpendicular with respect to Θ. Proof of Proposition 4.2. Let M be a decomposable Monge–Ampère system with a Lag- rangian pair (E1, E2) on (M,D). First we observe that M has a decomposable local generator. To see this, let X1, . . . , Xn and P1, . . . , Pn be local frames of E1 and E2 respectively. Let R be the Reeb vector field for a local contact form θ defining D. Consider the dual coframe θ, α1, . . . , αn, β1, . . . , βn of T ∗M to the frame R,X1, . . . , Xn, P1, . . . , Pn of TM . Then we see, by the decomposing condition, that there exist an (n − 1)-form γ and a non-vanishing function µ on M such that ω = µ(β1 ∧ · · · ∧ βn) + θ ∧ γ. Thus we have M = 〈β1 ∧ · · · ∧ βn, θ, dθ〉. Note that Ann(θ, α1, . . . , αn) = E1. Then, by Lemma 4.3, we see that M is determined just by E1. In particular, given a Lagrangian pair (E1, E2), the decomposable Monge–Ampère system with the Lagrangian pair (E1, E2) is uniquely determined. � Remark 4.4. If the characteristic system E1 is integrable, then the decomposable Monge– Ampère system is isomorphic to the system corresponding to the equation Hess = 0 (see [22, 24]). 14 G. Ishikawa and Y. Machida Remark 4.5. The equation Hess = 0 has the infinite-dimensional automorphism pseudo- group which consists of the lifts of diffeomorphisms on the dual projective space. However, if a Lagrangian pair is associated, then the automorphism pseudo-group turns to be of finite- dimensional, as stated in Proposition 4.2. 5 Automorphisms of Monge–Ampère systems with Lagrangian pairs Let us consider the automorphism pseudo-group Aut(M) of all local isomorphisms of a bi- decomposable Monge–Ampère systemM = 〈θ, dθ, ω〉 with a Lagrangian pair (E1, E2) on a con- tact manifold (M,D) of dimension 2n+ 1. By Theorems 3.2 and 3.8, in the case n ≥ 3, any automorphism of any bi-decomposable Monge–Ampère system with a Lagrangian pair (E1, E2) preserves the Lagrangian pair (E1, E2) up to the interchange of E1, E2. Therefore infinitesimal symmetries of bi-decomposable Monge– Ampère systems are studied based on infinitesimal symmetries of Lagrangian contact structures. Now let G be G =  c2 0 0 b1 cA On b2 On c tA−1 ∣∣∣∣∣∣ c ∈ R×, A ∈ SL(n,R), b1, b2 ∈ Rn  . Proposition 5.1. The bi-decomposable Monge–Ampère systems with Lagrangian pairs are in bijective correspondence with G-structures of type m. Proof. We consider, as in Section 4, the fundamental GLA of contact type m = g−2 ⊕ g−1 and the decomposition g−1 = e1 ⊕ e2 into the Lagrangian pair. Moreover we fix a volume form Ω1 ∈ ∧n(e1∗) on e1 and a volume form Ω2 ∈ ∧n(e2∗) on e2. Consider the group C(m; e1,Ω1; e2,Ω2) consisting of all a ∈ GL(m) which satisfies the following conditions: ag−1 = g−1, the graded linear automorphism a of m induced by a is a GLA-automorphism, ae1 = e1, ae2 = e2, and a∗Ω1 = λΩ1, a∗Ω2 = λΩ2 for some λ ∈ R×. Then we have that C(m; e1,Ω1; e2,Ω2) is identical with G. � Thus the equivalence problem of bi-decomposable Monge–Ampère systems with Lagrangian pairs is studied as an adapted G-structure on a contact manifold of dimension 2n+ 1. Let G0 be a subgroup of GL(n+ 2,R) defined by G0 =   k−1 0 0 b1 A 0 a tb2 k ∣∣∣∣∣∣ k ∈ R×, A ∈ SL(n,R), b1, b2 ∈ Rn, a ∈ R  . We set H0 =   k−1 0 0 0 A 0 0 0 k ∣∣∣∣∣∣ k ∈ R×, A ∈ SL(n,R)  ⊂ G0. Then we have the model space G0/H0 ∼= R2n+1 with coordinates (x, z, p). The G0-action on R2n+1 is described by (x, z, p) 7→ (x′, z′, p′) = ( k(Ax+ b1), k ( kz − tb2x− a ) , k ( p− 1 k tb2 ) A−1 ) . Then dz′−p′dx′ = k2(dz−pdx) and the form ω = cdx1∧· · ·∧dxn−dp1∧· · ·∧dpn is transformed to ω′ = knω. Monge–Ampère Systems with Lagrangian Pairs 15 Let Ei,j , 0 ≤ i, j ≤ n+ 1, denotes the elementary (n+ 2)× (n+ 2)-matrix such that the (i, j) component is 1 and other components are all zero. We set ε = −E0,0 + En+1,n+1, ei = Ei,0 (1 ≤ i ≤ n), fj = En+1,j (1 ≤ j ≤ n), γ = En+1,0, and identify the set of traceless matrices sl(n,R) with 0 0 0 0 A 0 0 0 0 ∣∣∣∣∣∣ A ∈ sl(n,R)  ⊂ g0. Then we set g0 = 〈ε〉R ⊕ sl(n,R), g1 −1 = 〈e1, . . . , en〉R, g2 −1 = 〈f1, . . . , fn〉R, g−1 = g1 −1 ⊕ g2 −1, and g−2 = 〈γ〉R. Then g0 = g−2 ⊕ g−1 ⊕ g0 is the Lie algebra of G0. We write m = g−2⊕ g−1. Then the Lie subalgebra m becomes the split Heisenberg algebra. The matrix representation of A0 ∈ g0 = 〈ε〉R ⊕ sl(n,R) with respect to the basis {γ, ei (1 ≤ i ≤ n), fj (1 ≤ i ≤ n)} in the Heisenberg algebra V = m = g−2 ⊕ g−1 has the following form: A0 = C +A′0 = c 2 0 0 0 I O 0 O I + 0 0 0 0 A O 0 O −tA  . In fact, for the commutators, we have by the direct calculations: [ε, γ] = 2γ, [ε, ei] = ei, [ε, fj ] = fj , [A, γ] = O, [A, ei] = ai, [A, fj ] = −aj , where ai is the i-th column of A and aj is the j-th row of A, 1 ≤ i, j ≤ n, and A ∈ sl(n,R). Moreover we have [ei, fj ] = −δijγ, [ei, ej ] = 0, [fi, fj ] = 0 (1 ≤ i, j ≤ n). We will study the prolongation of (m, g0). We define the prolongation inductively gk = g(m, g0)k (k ≥ 1) by the set of elements {(α, β) ∈ Hom(g−1, gk−1)⊕Hom(g−2, gk−2)} satisfying (i) β([x, y]) = [α(x), y]− [α(y), x] (x, y ∈ g−1), (ii) [α(y), z] = [β(z), y] (y ∈ g−1, z ∈ g−2). See [31, p. 429]. Then we have Lemma 5.2. The prolongation gi vanishes for any i ≥ 1. Proof. First we calculate g1. For any (α, β) ∈ Hom(g−1, g0)⊕Hom(g−2, g−1), the conditions (i) and (ii) imply that (i) β([ei, fj ]) = [α(ei), fj ]− [α(fj), ei] (1 ≤ i, j ≤ n), (ii-1) [α(ei), γ] = [β(γ), ei] (1 ≤ i ≤ n), (ii-2) [α(fj), γ] = [β(γ), fj ] (1 ≤ j ≤ n). 16 G. Ishikawa and Y. Machida Set β(γ) = n∑ `=1 b`e` + n∑ m=1 cmfm, α(ei) = hiε+A(i), α(fj) = kjε+B(j), where b`, cm, hi, kj ∈ R, A(i), B(j) ∈ sl(n,R). Then β([ei, fj ]) = n∑ `=1 (−δijb`)e` + n∑ m=1 (−δijcm)fm, [α(ei), fj ] = hifj − ( j-th row of A(i) ) , [α(fj), ei] = kjei + ( i-th column of B(j) ) , [α(ei), γ] = 2hiγ, [β(γ), ei] = ciγ, [α(fj), γ] = 2kjγ, [β(γ), fj ] = −bjγ. By the condition (i), we have B (j) `i = −δ`ikj + δijb`, A (i) jm = δjmhi + δijcm (1 ≤ i, j, `,m ≤ n). By the condition (ii-1), we have ci = 2hi, 1 ≤ i ≤ n. By the condition (ii-2), we have bj = −2kj , 1 ≤ j ≤ n. Therefore we have B (j) `i = −δ`ikj − 2δijk`, A (i) jm = δjmhi + 2δijhm (1 ≤ i, j, `,m ≤ n). In particular, for any j, ` with 1 ≤ j, ` ≤ n, we have B (j) `` = −kj − 2δ`jk`, and therefore 0 = trB(j) = −(n+ 2)kj , hence kj = 0, 1 ≤ j ≤ n. For any i, m with 1 ≤ i,m ≤ n, we have A (j) mm = hi + 2δimhm, and therefore 0 = trA(i) = (n+ 2)hi, hence hi = 0, 1 ≤ i ≤ n. Thus we have β = 0 and α = 0. Therefore we have g1 = 0. Second we calculate g2. Take any (α, β) ∈ g2 ⊂ Hom(g−1, g1)⊕ Hom(g−2, g0). Since g1 = 0, α = 0. Then we see β(γ) = 0. Therefore β = 0. Thus we obtain that g2 = 0. From g1 = 0, g2 = 0, we have gi = 0, i ≥ 3, automatically. � Then we have Theorem 5.3. Let (M,D) be a contact manifold of dimension 2n+1 andM a bi-decomposable Monge–Ampère system with a Lagrangian pair on (M,D). Assume that n ≥ 3. Then the automorphism pseudo-group Aut(M) ofM has at most the dimension of (n+1)2. The estimate is best possible. Proof. By Lemma 5.2, we have known that the prolongations gi, 1 ≤ i in the sense of Tanaka vanish. On the other hand G is equal to (H0)# in the notation of [29]. Then, by the finiteness theorem of Tanaka [29, Corollary 2], we see that the dimension of the pseudo-group of auto- morphisms on M is estimated by dim(g0) = ∑ i≤0 dim(gi) = (n + 1)2. Moreover there exists an Monge–Ampère system M with a Lagrangian pair on M = R2n+1, which arises in equi-affine geometry, such that the automorphism group Aut(M) attains the maximal dimension (n+ 1)2. See Section 8.1. � Remark 5.4. The sharp symmetry bounds of non-flat Lagrangian contact structures together with many other parabolic geometries have been obtained by Kruglikov and The [18]. Lemma 5.2 in our paper is similar to the concept of prolongation rigidity studied by them in [18]. Monge–Ampère Systems with Lagrangian Pairs 17 6 Hesse representations Let (E1, E2) be a Lagrangian pair on a contact manifold (M,D). We call (E1, E2) bi-Legendre- integrable or simply integrable if both E1 and E2 are completely integrable as subbundles in TM . Then (E1, E2) defines a pair of Legendrian foliations (E1, E2) on M locally. The standard Lagrangian pair (Est 1 , E st 2 ) introduced in Section 2 is integrable. In fact the foliation Est 1 is defined by fibers of the projection (x, z, p) 7→ ( p, n∑ i=1 xipi − z ) and the foliation Est 2 is defined by (x, z, p) 7→ (x, z). Definition 6.1. If a Lagrangian pair (E1, E2) on (M,D) is locally contactomorphic to the standard Lagrangian pair (Est 1 , E st 2 ) on (R2n+1, Dst), then we call (E1, E2) flat. Let (E1, E2) be a Lagrangian pair on a contact manifold (M,D) of dimension 2n + 1. As- sume that (E1, E2) is bi-Legendre-integrable. Then, locally, there exist Legendrian fibrations π1 : M →W1 and π2 : M →W2 having E2 and E1 as the kernels of the differentials (π1)∗ : TM → TW1 and (π2)∗ : TM → TW2 for some manifolds W1 and W2 of dimension n + 1 respectively. Then we have the following diagram: M π1 }} π2 !! W1 W2 We call (π1, π2) a double Legendrian fibration. In this case we say that W1 and W2 are in the dual relation via the Legendre transformation on M . Since D = E1 ⊕E2, we see (π1, π2) : M → W1×W2 is an immersion. Thus M has a pseudo-product structure doubly foliated by Legendrian submanifolds [27, 30]. Example 6.2. The projective cotangent bundle M = P (T ∗W ) over a manifold W of dimension n + 1 with the canonical projection π1 : M → W has the canonical contact structure D ⊂ TM defined by, for any (x, [α]) ∈M with x ∈W , [α] ∈ P (T ∗xW ), D(x,[α]) = {v ∈ TxM \|α((π1)∗v) = 0}. We see moreover that D has the integrable Lagrangian subbundle E2 = Ker(π1∗). Assume that W is the projective space Pn+1. Then the contact manifold M = P (T ∗Pn+1) is naturally identified with the contact manifold P (T ∗Pn+1∗), the projective cotangent bundle over the dual projective space Pn+1∗ with the canonical projection π2 : P (T ∗Pn+1∗) → Pn+1∗ [13]. Then D ⊂ TM has another Lagrangian subbundle E1 = Ker((π2)∗) and (E1, E2) turns to be an integrable Lagrangian pair of (M,D). There is associated to (E1, E2) the double Legendrian fibration P ( T ∗Pn+1 ) π1 xx π2 && Pn+1 Pn+1∗ for W1 = Pn+1, W2 = Pn+1∗. This globally defined Lagrangian pair (E1, E2) is flat. In fact, P (T ∗Pn+1) is identified with the incidence hypersurface I ⊂ Pn+1 × Pn+1∗ defined by x0y0 + x1y1 + · · ·+ xn+1yn+1 = 0 18 G. Ishikawa and Y. Machida for homogeneous coordinates [x] = [x0 : x1 : · · · : xn+1] of Pn+1 and [y] = [y0 : y1 : · · · : yn+1] of Pn+1∗. On the affine open subset U = {x0 6= 0, yn+1 6= 0} ⊂ Pn+1 × Pn+1∗, take local coordinates of x′i = xi x0 , z′ = −xn+1 x0 , p′j = yj yn+1 , z̃′ = − y0 yn+1 , (1 ≤ i ≤ n, 1 ≤ j ≤ n). Then I ∩ U is defined by −z′ − z̃′ + n∑ i=1 x′ip ′ i = 0, where we have dz′− n∑ i=1 p′idx ′ i+ ( dz̃′− n∑ i=1 x′idp ′ i ) = 0. The contact structure on I∩U is given by dz′− n∑ i=1 p′idx ′ i = 0. Also it is given by dz̃′ − n∑ i=1 x′idp ′ i = 0. This shows (E1, E2)\|U is flat. Also on other affine open subsets, we can verify the flatness of (E1, E2) similarly or by an argument using homogeneity. Example 6.3. For a Riemannian manifold W = (W, g) of dimension n + 1, the unit tangent bundle M = T1W of W has the canonical contact structure D ⊂ TM , D(x,v) = { u ∈ T(x,v)M \|π∗u ∈ v⊥ } , (x, v) ∈ T1W, and the canonical Lagrangian pair (E1, E2) induced by the horizontal lift and the vertical lift of the Levi-Civita connection: (E1)(x,v) = ( v⊥ )hor , (E2)(x,v) = ( v⊥ )ver , (x, v) ∈ T1W. Here π : M −→W is the canonical projection and v⊥ = {w ∈ TxW \| g(v, w) = 0}. Then (E1, E2) is bi-Legendre-integrable if and only if W is a space form. In fact, the vertical lift E2 is always completely integrable. Moreover the horizontal lift E1 is completely integrable if and only if W is projectively flat, that is, a space form (see [28, Corollaries 3.5 and 6.5]). The bi-decomposable class of Monge–Ampère systems with flat Lagrangian pairs turns out to be an intrinsic representation of the well-known class of Monge–Ampère equations locally expressed by Hesse representations: Hess(z) = F (x1, . . . , xn, z, p1, . . . , pn) (6= 0). Thus we are led to the following definition: Definition 6.4. We call a Monge–Ampère system with a flat Lagrangian pair a Hesse Monge– Ampère system. Note that there are sub-classes of Hesse Monge–Ampère equations, Euler–Lagrange Monge– Ampère equations: Hess(z) = F1(x1, . . . , xn, z) · F2 p1, . . . , pn, n∑ j=1 pjxj − z  ( 6= 0), (see [5, p. 21, Example 2]) and flat Monge–Ampère equations: Hess(z) = c (c is constant, c 6= 0). Definition 6.5. LetM be a Hesse Monge–Ampère system. Then we callM an Euler–Lagrange Monge–Ampère system if M is locally generated by a bi-decomposable n-form ω = ω1 − ω2 satisfying dω1 ≡ 0, dω2 ≡ 0 mod θ Monge–Ampère Systems with Lagrangian Pairs 19 Thus we have a sequence of classes of Monge–Ampère systems: {M-A system with Lagrangian pair} ⊃ {M-A system with integrable Lag. pair} ⊃ {Hesse M-A system} ⊃ {Euler–Lagrange M-A system} ⊃ {flat M-A system}. We will study Hesse Monge–Ampère systems in detail, and, in fact, we characterize the above three classes in intrinsic way. Note that the equation of non-zero constant Gauss–Kronecker curvature K = c falls into the class of Euler–Lagrange Monge–Ampère systems and the equation of improper affine hyperspheres Hess = c falls into the class of flat Monge–Ampère systems. We will show, in Sections 8.2–8.5, that Monge–Ampère systems defined by the Gaussian curvature constant equation K = c in En+1, Sn+1, Hn+1 are Euler–Lagrange systems. Proposition 6.6. LetM be a Hesse Monge–Ampère system, i.e., a Monge–Ampère system with a flat Lagrangian pair (E1, E2) generated by an n-form ω = ω1−ω2 enjoying the bi-decomposing condition. Then M is locally isomorphic to a Monge–Ampère system M′ on an open subset U ⊂ R2n+1 with the standard Lagrangian pair (Est 1 , E st 2 ) which is locally generated by an n-form of type F (x1, . . . , xn, z, p1, . . . , pn)dx1 ∧ · · · ∧ dxn − dp1 ∧ · · · ∧ dpn for a non-vanishing function F on U . In particular, there exists a system of local Darboux coor- dinates (x1, . . . , xn, z, p1, . . . , pn) in some neighborhood of each point, such thatM is represented by a Hesse Monge–Ampère equation of the form Hess(z) = F (x1, . . . , xn, z, p1, . . . , pn) (F 6= 0). Proof. Since (E1, E2) is flat, around each point of M , there exists a system of local coordinates (x1, . . . , xn, z, p1, . . . , pn) such that D = {θ = 0}, θ = dz − n∑ i=1 pidxi, and that E1 = Ker(π2∗) = 〈 ∂ ∂x1 + p1 ∂ ∂z , . . . , ∂ ∂xn + pn ∂ ∂z 〉 , E2 = Ker(π1∗) = 〈 ∂ ∂p1 , . . . , ∂ ∂pn 〉 , π1 : R2n+1 → Rn+1, π1(x1, . . . , xn, z, p1, . . . , pn) = (x1, . . . , xn, z), π2 : R2n+1 → Rn+1, π2(x1, . . . , xn, z, p1, . . . , pn) = ( p1, . . . , pn, n∑ i=1 pixi − z ) . This means that M is locally isomorphic to a Monge–Ampère system with the standard Lag- rangian pair. Then, from the bi-decomposing condition, we have, setting x = (x1, . . . , xn), p = (p1, . . . , pn), ω = ω1 − ω2 = f(x, z, p)dx1 ∧ · · · ∧ dxn − g(x, z, p)dp1 ∧ · · · ∧ dpn for some functions f(x, z, p), g(x, z, p) ( 6= 0) on R2n+1. Therefore, putting F = f/g, we have a form Hess(z) = F (x, z, p). � Proposition 6.7. Let M be a Hesse Monge–Ampère system. Then M is an Euler–Lagrange Monge–Ampère system if and only if M is locally isomorphic to a Monge–Ampère system M′ on an open subset U ⊂ R2n+1 with the standard Lagrangian pair generated by an n-form of type F1(x1, . . . , xn, z) · F2(z̃, p1, . . . , pn)dx1 ∧ · · · ∧ dxn − dp1 ∧ · · · ∧ dpn 20 G. Ishikawa and Y. Machida for some non-vanishing functions F1 of x, z and F2 of z̃ = n∑ j=1 pjxj − z and p. In particular, M is locally represented as Hess(z) = F1(x1, . . . , xn, z) · F2(z̃, p1, . . . , pn) (F1, F2 6= 0). Proof. By the assumption, we can set ω1 = f(x, z, p)dx1 ∧ · · · ∧ dxn, ω2 = g(x, z, p)dp1 ∧ · · · ∧ dpn. Since dω1 = df ∧ dx1 ∧ · · · ∧ dxn = ( n∑ i=1 ∂f ∂xi dxi + ∂f ∂z dz + n∑ i=1 ∂f ∂pi dpi ) ∧ dx1 ∧ · · · ∧ dxn ≡ n∑ i=1 ∂f ∂pi dpi ∧ dx1 ∧ · · · ∧ dxn, mod θ, we see dω1 ≡ 0 mod θ if and only if f is independent of p1, . . . , pn: f = f(x1, . . . , xn, z). Besides, we take another system of local coordinates (x, z̃, p) with z̃ = n∑ i=1 xipi − z. Then θ = − ( dz̃ − n∑ i=1 xidpi ) and we have dω2 = dg ∧ dp1 ∧ · · · ∧ dpn = ( n∑ i=1 ∂g ∂xi dxi + ∂g ∂z̃ dz̃ + n∑ i=1 ∂g ∂pi dpi ) ∧ dp1 ∧ · · · ∧ dpn ≡ n∑ i=1 ∂g ∂xi dxi ∧ dp1 ∧ · · · ∧ dpn, mod θ. Therefore we see dω2 ≡ 0 mod θ if and only if g is independent of x for the system of local coordinates (x, z̃, p): g = g(z̃, p1, . . . , pn). Therefore putting F1 = f , F2 = 1/g, we have a form Hess(z) = F1(x, z) · F2(z̃, p). � By Theorem 3.8, we have the following: Proposition 6.8. Let n ≥ 3. Then Definition 6.4 (resp. Definition 6.5) depends only on the Monge–Ampère system and does not depend on the choice of Lagrangian pairs of the Monge– Ampère system. The class of Hesse Monge–Ampère systems (resp. the class of Euler–Lagrange Monge–Ampère systems) is invariant under contact transformations. Proof. Let M be a Monge–Ampère system with a Lagrangian pair on a contact manifold (M,D). Let (E1, E2) and (E′1, E ′ 2) be two Lagrangian pairs associated to M. Then, by Theo- rem 3.8, E′1 = E1, E′2 = E2 or E′1 = E2, E′2 = E1. Therefore the flatness of Lagrangian pair depends only onM. Moreover it is clear that, the condition of Definition 6.5, i.e., the possibility of a bi-decomposition ω = ω1−ω2 of a local generator ω into closed decomposable forms ω1, ω2 up to a contact form θ depends only on M. Let Φ: (M,D) → (M ′, D′) be a contact transformation between (M,D) and another con- tact manifold (M ′, D′). Set M′ = Φ−1∗M. Then M′ is a Monge–Ampère system with the Lagrangian pair (Φ∗E1,Φ∗E2). If (E1, E2) is flat, then so is (Φ∗E1,Φ∗E2). Moreover if ω = ω1 − ω2 is a bi-decomposition satisfying the condition of Definition 6.5 for M, then (Φ−1)∗ω = (Φ−1)∗ω1 − (Φ−1)∗ω2 satisfies the condition of Definition 6.5 for M′. � Monge–Ampère Systems with Lagrangian Pairs 21 Remark 6.9. The Euler–Lagrange systems are studied in [5] via the key notion “Poincaré– Cartan form”. If an Euler–Lagrange Monge–Ampère system with Lagrangian pair is given by θ = dz − n∑ i=1 pidxi = − ( dz̃ − n∑ i=1 xidpi ) , ω = f(x, z)dx1 ∧ · · · ∧ dxn − g(p, z̃)dp1 ∧ · · · ∧ dpn, then the Poincaré–Cartan form is given by Π = θ ∧ ω = f(x, z)dz ∧ dx1 ∧ · · · ∧ dxn − g(p, z̃)dz̃ ∧ dp1 ∧ · · · ∧ dpn. To conclude this section, we characterize, among others, the class of equations Hess = c in term of the projective structure: Proposition 6.10. Let M be an Euler–Lagrange Monge–Ampère system on M = P (T ∗Pn+1) induced by the diagram P ( T ∗Pn+1 ) π1 xx π2 && Pn+1 Pn+1∗ and W1 = Pn+1, W2 = Pn+1∗, generated by a bi-decomposable n-form ω = ω1 − ω2. Then the condition ∇ω1 = 0, ∇ω2 = 0 is satisfied for the covariant derivative ∇ of the flat connection induced on each local projective chart of W1 = Pn+1 if and only if M is represented by a Monge–Ampère equation Hess(z) = c (c is constant, c 6= 0). Proof. From the equivalence between the conditions ∇ω1 = 0 and df = 0 (resp. ∇ω2 = 0 and dg = 0) in the proof of Proposition 6.6, we have that f , g are non-zero constants. Therefore we have the form Hess(z) = c (6= 0). � 7 A method to construct Monge–Ampère systems with Lagrangian pairs Let (M,D) be a contact manifold of dimension 2n+1 and (E1, E2) a Lagrangian pair on (M,D). Consider the quotient bundle TM/E2 (resp. TM/E1) of rank n+ 1 and a section ω′1 (resp. ω′2) to the line bundle ∧n+1(TM/E2)∗ (resp. ∧n+1(TM/E1)∗) off the zero-section. Let θ be a local contact form defining D. Recall that the Reeb vector field R = Rθ is defined by the condition iRθθ = 1, iRθdθ = 0. Then we define an n-form on M by ω = iRθ ( Π∗1ω ′ 1 −Π∗2ω ′ 2 ) . Here Π1 : TM → TM/E2 (resp. Π2 : TM → TM/E1) denotes the bundle projection, and Π∗1 : (TM/E2)∗ → T ∗M and Π∗1 : ∧n+1 (TM/E2)∗ → ∧n+1T ∗M (resp. Π∗2 : (TM/E1)∗ → T ∗M and Π∗2 : ∧n+1 (TM/E1)∗ → ∧n+1T ∗M) its dual injections. Then we have the following basic lemma for our construction: 22 G. Ishikawa and Y. Machida Lemma 7.1. The differential system M = 〈θ, dθ, ω〉, generated by ω and the contact form θ, is independent of the choice of θ, and depends only on given ω′1, ω′2. We call M the Monge–Ampère system with (E1, E2) induced from ω′1, ω′2. To show Lemma 7.1, take a local symplectic frame X1, . . . , Xn, P1, . . . , Pn of D with respect to dθ: dθ(Xi, Xj) = 0, dθ(Pi, Pj) = 0, dθ(Xi, Pj) = δij , with E1 = 〈X1, . . . , Xn〉, E2 = 〈P1, . . . , Pn〉. Then X1, . . . , Xn, P1, . . . , Pn, Rθ form a local frame of TM . Lemma 7.2. The Reeb vector field Rθ′ for a contact form θ′ = ρθ defining D is given by Rθ′ = 1 ρ2 [ n∑ i=1 (Piρ)Xi − n∑ i=1 (Xiρ)Pi + ρRθ ] . Proof. Let α1, . . . , αn, β1, . . . , βn, θ be the frame of T ∗M dual to X1, . . . , Xn, P1, . . . , Pn, Rθ. Set Rθ′ = ∑ i aiXi + ∑ j bjPj + cRθ. Then, by iRθ′θ ′ = 1, we have c = 1 ρ . Besides we have iRθ′dθ ′ = [iRθ′dρ]θ − iRθ′θ dρ+ ρiRθ′dθ = ∑ i ai(Xiρ) + ∑ j bj(Pjρ) + c(Rθρ)  θ − cdρ+ ρ ∑ i aiβi − ∑ j bjαj  , while dρ = ∑ i(Xiρ)αi + ∑ j(Pjρ)βj + (Rθρ)θ. Therefore, by iRθ′dθ ′ = 0, we have − ∑ i [ρbi + c(Xiρ)]αi + ∑ j [ρaj − c(Pjρ)]βj + ∑ i ai(Xiρ) + ∑ j bj(Pjρ)  θ = 0. Thus we have ai = 1 ρ2 Piρ, bi = − 1 ρ2 Xiρ, c = 1 ρ . � Proof of Lemma 7.1. We set Π∗1ω ′ 1 = λ ·θ∧α1∧· · ·∧αn for a function λ. Then iRθ(Π ∗ 1ω ′ 1) = λ · α1 ∧ · · · ∧ αn. By Lemma 7.2, we have for θ′ iRθ′ ( Π∗1ω ′ 1 ) ≡ 1 ρ λ · α1 ∧ · · · ∧ αn = 1 ρ · iRθω ′ 1, mod θ. Similarly we have iRθ′ (Π ∗ 2ω ′ 2) ≡ 1 ρ · iRθω ′ 2, mod θ. Therefore iRθ′ ( Π∗1ω ′ 1 −Π∗2ω ′ 2 ) ≡ 1 ρ iRθ ( Π∗1ω ′ 1 −Π∗2ω ′ 2 ) , mod θ. Therefore M = 〈θ, dθ, ω〉 is independent of the choice of θ. � Monge–Ampère Systems with Lagrangian Pairs 23 Suppose that (E1, E2) is integrable. Let M2n+1 π1 zz π2 $$ Wn+1 1 Wn+1 2 be a double Legendrian fibration induced by a Lagrangian pair (E1, E2) locally. Suppose that a volume (n + 1)-form Ω1 is given on W1 (resp. a volume (n + 1)-form Ω2 on W2). Since (TM/E2)x ∼= Tπ1(x)W1 via π1 (resp. (TM/E1)x ∼= Tπ2(x)W2 via π2) for x ∈ M , Ω1 (resp. Ω2) is regarded as a non-zero section ω′1 of ∧n+1(TM/E1)∗ (resp. ω′2 of ∧n+1(TM/E2)∗). Then, following the general setting, we set ω = iRθ ( π∗1Ω1 − π∗2Ω2 ) . Note that π∗1Ω1, π∗2Ω2 are basic forms (cf. [14]) for π1, π2 respectively, however iRθπ ∗ 1Ω1 and iRθπ ∗ 2Ω2 need not to be basic. Example 7.3. Consider R2n+1 with coordinates (x, z, p) = (x1, . . . , xn, z, p1, . . . , pn), D = {θ = 0}, θ = dz − n∑ i=1 pidxi, E st 1 = 〈 ∂ ∂x1 + p1 ∂ ∂z , . . . , ∂ ∂xn + pn ∂ ∂z 〉, E st 2 = 〈 ∂∂p1 , . . . , ∂ ∂pn 〉, π1 : R2n+1 → Rn+1, π1(x, z, p) = (x, z), π2 : R2n+1 → Rn+1, π2(x, z, p) = (p, x · p− z). Set z̃ = x · p− z. The Reeb vector field for θ is given by R = ∂ ∂z . Let π∗1ω ′ 1 = f(x, z, p)dz ∧ dx1 ∧ · · · ∧ dxn be a non-zero section of ∧n+1(TR2n+1/E2)∗ pulled- back to an (n + 1)-form on R2n+1. Then iR(π∗1ω ′ 1) = f(x, z, p)dx1 ∧ · · · ∧ dxn. Similarly, let π∗2ω ′ 2 = −g(x, z, p)dz̃ ∧ dp1 ∧ · · · ∧ dpn be a non-zero section of ∧n+1(TR2n+1/E1)∗ pulled-back to an (n + 1)-form on R2n+1. Then iR(π∗2ω ′ 2) = g(x, z, p)dp1 ∧ · · · ∧ dpn. Thus, following the general setting, we have ω = f(x, z, p)dx1 ∧ · · · ∧ dxn − g(x, z, p)dp1 ∧ · · · ∧ dpn, and we obtain Hesse Monge–Ampère systems. Further, let Ω1 = f(x, z)dz ∧ dx1 ∧ · · · ∧ dxn (resp. Ω2 = −g(p, z̃)dz̃ ∧ dp1 ∧ · · · ∧ dpn) be a volume form on W1 = Rn+1 (resp. on W2 = Rn+1). Then iRπ ∗ 1Ω1 = f(x, z)dx1 ∧ · · · ∧ dxn (resp. iRπ ∗ 2Ω2 = g(p, z̃)dp1 ∧ · · · ∧ dpn). Thus we have ω = f(x, z)dx1 ∧ · · · ∧ dxn − g(p, z̃)dp1 ∧ · · · ∧ dpn, and we obtain Euler–Lagrange Monge–Ampère systems. Remark 7.4. The Poincaré–Cartan form is given by Π = π∗1Ω1 − π∗2Ω2 (see Remark 6.9). 8 Homogeneous Monge–Ampère systems with Lagrangian pairs 8.1. Monge–Ampère system on R2n+1 as Hess = c on Rn+1. The Monge–Ampère system with Lagrangian pair for the equation Hess(f) = c in equi-affine geometry is given as follows. Consider the contact manifold M = R2n+1 with coordinates (x, z, p) = (x1, . . . , xn, z, p1, . . . , pn) and with the contact form θ = dz− n∑ i=1 pidxi. We set two Lagrangian sub-bundles E1, E2 of the contact distribution D = {θ = 0} by E1 = 〈 ∂ ∂x1 + p1 ∂ ∂z , . . . , ∂ ∂xn + pn ∂ ∂z 〉 , E2 = 〈 ∂ ∂p1 , . . . , ∂ ∂pn 〉 , 24 G. Ishikawa and Y. Machida which form a Lagrangian pair of (M,D). The Reeb vector field R is given by ∂ ∂z . Note that −θ = d ( n∑ i=1 pixi − z ) − n∑ i=1 xidpi. Then we have the double Legendrian fibration induced by (E1, E2): M = R2n+1 π1 vv π2 (( W1 = Rn+1 W2 = Rn+1 where π1(x, z, p) = (x, z), π2(x, z, p) = (p, z̃), z̃ = n∑ i=1 pixi − z. Moreover take the (n+ 1)-forms Ω1 = c(dz ∧ dx1 ∧ · · · ∧ dxn), Ω2 = −dz̃ ∧ dp1 ∧ · · · ∧ dpn on W1 = Rn+1 (c ∈ R, c 6= 0) and on W2 = Rn+1 respectively. Then ω = iR(π∗1Ω1 − π∗2Ω2) is given by ω = cdx1 ∧ · · · ∧ dxn − dp1 ∧ · · · ∧ dpn. Thus we construct the Monge–Ampère system M = 〈θ, dθ, ω〉 with the Lagrangian pair (E1, E2) globally on M = R2n+1. Proposition 8.1. Under the situation above, we have a Monge–Ampère system M generated by (θ, dθ, ω) with a Lagrangian pair (E1, E2) on M = R2n+1. The projection to W1 = Rn+1 of a geometric solution of M satisfies the equation Hess(f) = c, when it is represented as a graph z = f(x1, . . . , xn) outside of its singular locus. The projection to W2 = Rn+1 of a geometric solution of M satisfies the equation Hess(f) = 1 c , when it is represented as a graph z′ = n∑ i=1 pixi − z = f(p1, . . . , pn) outside of its singular locus. Remark 8.2. The Monge–Ampère system for Hess = c is isomorphic to the system for Hess = 1 if c > 0, and is isomorphic to the system for Hess = −1 if c < 0. Remark 8.3. The Monge–Ampère system for Hess = c, c 6= 0 has the natural symmetry by the group G′ of equi-affine transformations on W1 = Rn+1 preserving the vector field ∂ ∂z . The group G′ is given by the semi-direct product G′ = G′′ n Rn+1 of G′′ ⊂ SL(n+ 1,R) and Rn+1, where G′′ = {( A 0 ta 1 ) ∣∣∣∣ A ∈ SL(n,R), a ∈ Rn } . Note that dimG′ = n(n+ 2) and also that each element of G′ is identified with1 0 0 b A 0 c ta 1  via an appropriate embedding SL(n + 1,R) ↪→ GL(n + 2,R). The Monge–Ampère system for Hess = c, c 6= 0, in fact, has bigger symmetry which attains the maximum for the dimension estimate of automorphisms given in Section 5 for bi-decomposable Monge–Ampère systems. Monge–Ampère Systems with Lagrangian Pairs 25 Let G be a subgroup of the projective transformation group PGL(n+2,R) on Rn+2 consisting of transformations represented by the matrices Ã = ` 0 0 b A 0 c ta k  , considered up to non-zero scalar multiples. Here `, k ∈ R×, A ∈ GL(n,R), a, b ∈ Rn, c ∈ R satisfying the condition (detA)2 = (k`)n. From the condition, we can take detA = ±1, ` = ±1/k in PGL(n + 2,R). If n is odd, then we can take detA = 1 and ` = 1/k. Note that dimG = (n+ 1)2. The group G acts on M = R2n+1 = Rn ×R×Rn transitively by Ã(x, z, p) = ( 1 ` (Ax+ b), 1 ` ( kz − tax− c ) , k ( p− 1 k ta ) A−1 ) . Here a, b, x are regarded as column n-vectors and p a row n-vector. The contact form θ = dz−pdx is transformed to k ` θ, therefore the contact structure D = {θ = 0} is G-invariant. The bi- decomposable form ω is transformed to 1 `n (detA)c(dx1 ∧ · · · ∧ dxn)− kn(detA)−1(dp1 ∧ · · · ∧ dpn) = 1 `n (detA)ω, by the condition (detA)2 = (k`)n. Therefore G leaves the Monge–Ampère systemM = 〈θ, dθ, ω〉 for Hess = c, c 6= 0. The group G acts on W1 = Rn+1 and on W2 = Rn+1 respectively by Ã(x, z) = ( 1 ` (Ax+ b), 1 ` ( kz − tax− c )) , and by Ã(z̃, p) = ( 1 ` ( kz̃ + k ( p− 1 k ta ) A−1b+ c ) , k ( p− 1 k ta ) A−1 ) . Then the volume form Ω1 on W1 (resp. Ω2 on W2) is transformed to k `n+1 (detA)Ω1 (resp. kn+1 ` (detA)−1Ω2). 8.2. Monge–Ampère system on T1E n+1 as K = c in En+1. In Sections 8.2–8.4, we describe the Monge–Ampère systems corresponding to the the equation of constant Gaussian curvature in Euclidean, spherical or hyperbolic geometry. To provide the concrete form of the Monge–Ampère system, we treat three cases separately. In the famous paper of Gauss [26], the “Gaussian curvature” of a space surface is introduced as the ratio of areas in the “Gauss map” of the surface. We observe that the equation of constant Gaussian curvature is regarded as a Monge–Ampère system with Lagrangian pair as follows. Consider the unit tangent bundle T1E n+1 = En+1 × Sn of the Euclidean space En+1. The standard contact structure on En+1 × Sn is given by the one-form θ = y1dx1 + y2dx2 + · · · + yn+1dxn+1 on En+1 ×En+1, restricted to En+1 × Sn. Here (x; y) = (x1, x2, . . . , xn+1; y1, y2, . . . , yn+1) is the system of coordinates on En+1 × En+1. We set the contact distribution D = {θ = 0} ⊂ T (En+1 × Sn) and two Lagrangian subbundles of D: E1 = { u = ξ1 ∂ ∂x1 + ξ2 ∂ ∂x2 + · · ·+ ξn+1 ∂ ∂xn+1 ∣∣∣ ξ1y1 + ξ2y2 + · · ·+ ξn+1yn+1 = 0 } , E2 = { v = η1 ∂ ∂y1 + η2 ∂ ∂y2 + · · ·+ ηn+1 ∂ ∂yn+1 ∣∣∣ v is tangent to Sn } , 26 G. Ishikawa and Y. Machida which form an integrable Lagrangian pair of (M,D). Then we have the double Legendrian fibration induced by (E1, E2): M = En+1 × Sn π1 vv π2 )) W1 = En+1 W2 = R× Sn Here π1(x, y) = x, π2(x, y) = (x · y, y) for (x, y) ∈ En+1 × Sn ⊂ En+1 ×En+1. Lemma 8.4. The above Lagrangian pair is f lat, namely, the bi-Legendrian f ibration is contac- tomorphic to the standard one: The Lagrangian pair (E1, E2) = (Ker(π2)∗,Ker(π1)∗) is flat (see the standard example in Introduction). Proof. On the open sets U = {(x, y) ∈ En+1 × Sn \| yn+1 6= 0} of En+1 × Sn and V = {(w, y) ∈ R × Sn \| yn+1 6= 0} of R × Sn, take the system of coordinates x′i = xi, p ′ i = − yi yn+1 , z′ = xn+1, 1 ≤ i ≤ n, on U , and define the diffeomorphism Φ: U → R2n+1 by Φ(x, y) = (x′, z′, p′). Moreover take the system of coordinates z̃′ = w yn+1 , p′i = − yi yn+1 , 1 ≤ i ≤ n, on V and define the diffeomorphism ψ : V → Rn+1 by ψ(w, y) = (z̃′, p′). We denote by ϕ : En+1 → Rn+1 the identity map, forgetting the Euclidean metric. Then (Φ, ϕ, ψ) induces the contactomorphism from (U,D;E1, E2) to (R2n+1, Dst;E st 1 , E st 2 ). In fact θ = yn+1 ( dz′ − n∑ i=1 p′idx ′ i ) = yn+1(Φ−1)∗θst and yn+1 = ±1 /√ 1 + n∑ i=1 (p′i) 2. Similarly on each open set {yi 6= 0}, 1 ≤ i ≤ n + 1, we see the flatness of (E1, E2). � We endow En+1 with the standard volume form Ω1 = cdx1 ∧ dx2 ∧ · · · ∧ dxn+1, multiplied with a real constant c ( 6= 0). Moreover we endow (z; y1, y2, . . . , yn+1) ∈ R × Sn (⊂ R×En+1) with the standard volume form on R× Sn Ω2 = dz ∧ n+1∑ i=1 ( (−1)i+1yidy1 ∧ · · · ∧ d̆yi ∧ · · · ∧ dyn+1 ) \|R×Sn . The Reeb vector field R on En+1 × Sn is given by R = y1 ∂ ∂x1 + y2 ∂ ∂x2 + · · · + yn+1 ∂ ∂xn+1 , the “tautological” vector field. Then we set ω = iR(π∗1Ω1 − π∗2Ω2). Since iRπ ∗ 2Ω2 = iRd(x · y) ∧ n+1∑ i=1 ( (−1)i+1yidy1 ∧ · · · ∧ d̆yi ∧ · · · ∧ dyn+1 ) = n+1∑ i=1 ((−1)i+1yidy1 ∧ · · · ∧ d̆yi ∧ · · · ∧ dyn+1), we have ω = c(y1dx2 ∧ · · · ∧ dxn+1 − y2dx1 ∧ dx3 ∧ · · · ∧ dxn+1 + · · ·+ (−1)nyn+1dx1 ∧ · · · ∧ dxn) − (y1dy2 ∧ · · · ∧ dyn+1 − y2dy1 ∧ dy3 ∧ · · · ∧ dyn+1 + · · ·+ (−1)nyn+1dy1 ∧ · · · ∧ dyn) on En+1 × Sn. This is exactly the reincarnation of the equation K = c from the original definition due to Gauss. Monge–Ampère Systems with Lagrangian Pairs 27 Proposition 8.5. Under the situation above, we have a Monge–Ampère system M generated by (θ, dθ, ω) with a Lagrangian pair (E1, E2) on M = T1E n+1 = En+1 × Sn. The Gaussian curvature of the projection to W1 = En+1 of a geometric solution of M is equal to constant c outside of singular locus. Remark 8.6. The Monge–Ampère system for K = c is isomorphic to the system for K = 1 if c > 0, and is isomorphic to the system for K = −1 if c < 0. Remark 8.7. Let G be the Euclidean group on En+1. The group G acts on the orthonormal frame bundle transitively, therefore so on the unit tangent sphere bundle (the orthonormal 1- Stiefel bundle) En+1 × Sn of En+1. We fix the origin 0 in En+1 and identify En+1 with Rn+1, giving the isomorphism G ∼= O(n + 1) n Rn+1. The contact structure D = {θ = 0} and the contact form θ = n+1∑ i=1 yidxi on En+1 × Sn are G-invariant, and any G-invariant contact form is a non-zero multiple of θ. The group G acts on R× Sn by g(r, v) = (g(0) · g(v) + r, g(v)) = (b ·Av + r,Av), Here, for g ∈ G, we set g(v) = Av + b (A ∈ O(n+ 1), b ∈ Rn+1). Then we get the diagram: En+1 = G/H ′ ← En+1 × Sn = G/H → R× Sn = G/H ′′, for the isotropy groups H, H ′ and H ′′ satisfying H ′ ∼= O(n+ 1)←↩ H ∼= O(n) ↪→ H ′′ ∼= O(n) n Rn. The G-invariant volume forms on En+1 and R×Sn are unique up to non-zero constant. We can construct the Monge–Ampère systemM = 〈θ, dθ, ω〉 with the Lagrangian pair (E1, E2) globally on M = T1E n+1. 8.3. Monge–Ampère system on T1S n+1 as K = c in Sn+1. Consider En+2 × En+2 with coordinates x = (x0, x1, . . . , xn+1), y = (y0, y1, . . . , yn+1). Set x · y = ∑n+1 i=0 xiyi, the standard inner product. Consider T1S n+1 = { (x, y) ∈ En+2 ×En+2 \| \|x\| = 1, \|y\| = 1, x · y = 0 } , the unit tangent bundle of Sn+1. The standard contact structure D of T1S n+1 is defined by the contact form θ = n+1∑ i=0 yidxi. Then we have the double Legendrian fibration M = T1S n+1 π1 vv π2 (( W1 = Sn+1 W2 = Sn+1 where π1(x, y) = x, π2(x, y) = y. Lemma 8.8. The above bi-Legendrian fibration is contactomorphic to the standard model: The Lagrangian pair (E1, E2) = (Ker(π2)∗,Ker(π1)∗) is flat. 28 G. Ishikawa and Y. Machida Proof. For instance, on the open subset U = {x0 6= 0, yn+1 6= 0} of T1S n+1, consider the system of coordinates x′i = xi x0 , z′ = −xn+1 x0 , p′i = yi yn+1 , 1 ≤ i ≤ n. Moreover we set z̃′ = − y0 yn+1 . Then z̃′ + z′ − n∑ i=1 x′ip ′ i = 0 is satisfied. Moreover we have θ = −x0yn+1 ( dz′ − n∑ i=1 p′idx ′ i ) = x0yn+1 ( dz̃′ − n∑ i=1 x′idp ′ i ) . Thus we easily see that the diffeomorphism Φ: U → R2n+1 defined by Φ(x, y) = (x′, z′, p′) provides the required contactomorphism. � We set Ω1 = c iX(dx0 ∧ · · · ∧ dxn+1), the standard volume form on W1 = Sn+1 multiplied by c(6= 0) ∈ R, and set Ω2 = iY (dy0 ∧ · · · ∧ dyn+1), the standard volume form on W2 = Sn+1. Here X = n+1∑ i=0 xi ∂ ∂xi and Y = n+1∑ i=0 yi ∂ ∂yi . The Reeb vector field R on T1S n+1 is given by R = n+1∑ i=0 ( yi ∂ ∂xi −xi ∂∂yi ) . Then we have the following, which is the geometric form of the equation “Gaussian curvature = c”. Proposition 8.9. The associated Monge–Ampère system M with Lagrangian pair on M = T1S n+1 is generated by θ and ω = iR ( π∗1Ω1 − π∗2Ω2 ) = c ∑ 0≤j<i≤n+1 (−1)i+jxiyjdx0 ∧ · · · ∧ ˘dxj ∧ · · · ∧ ˘dxi ∧ · · · ∧ dxn+1 − c ∑ 0≤i<k≤n+1 (−1)i+kxiykdx0 ∧ · · · ∧ ˘dxi ∧ · · · ∧ ˘dxk ∧ · · · ∧ dxn+1 + ∑ 0≤j<i≤n+1 (−1)i+jyixjdy0 ∧ · · · ∧ ˘dyj ∧ · · · ∧ d̆yi ∧ · · · ∧ dyn+1 − ∑ 0≤i<k≤n+1 (−1)i+kyixkdy0 ∧ · · · ∧ d̆yi ∧ · · · ∧ ˘dyk ∧ · · · ∧ dyn+1. The Gaussian curvature of the projection to W1 = Sn+1 of a geometric solution ofM is equal to constant c outside of singular locus, while the Gaussian curvature of the projection to W2 = Sn+1 of a geometric solution of M is equal to constant 1 c outside of singular locus, Remark 8.10. Note that the Gaussian curvature K of a hypersurface in the unit sphere Sn+1 and its sectional curvature S as a Riemannian manifold are related by S = K+ 1. For example, the great sphere Sn ⊂ Sn+1 has the constant Gauss map to Sn+1 and K = 0, while has the sectional curvature 1. Remark 8.11. The group G = O(n+2) acts on W1 = Sn+1, on W2 = Sn+1 and on Sn+1×Sn+1, thus on T1S n+1 = {(x, y) ∈ Sn+1 × Sn+1 \|x · y = 0}. The contact structure D = {θ = 0} and the contact form θ = n+1∑ i=0 yidxi are G-invariant, and any contact form defining D is a non-zero constant multiple of θ. Monge–Ampère Systems with Lagrangian Pairs 29 We get the diagram: W1 = Sn+1 = G/H ′ ←M = T1S n+1 = G/H →W2 = Sn+1 = G/H ′′, for the isotropy groups H, H ′ and H ′′ satisfying H ′ ∼= O(n+ 1)←↩ H ∼= O(n) ↪→ H ′′ ∼= O(n+ 1). The G-invariant volume forms on W1 = Sn+1 and W2 = Sn+1 are unique up to non-zero constant. We can construct the Monge–Ampère system M = 〈θ, dθ, ω〉 with Lagrangian pair globally on M = T1S n+1. 8.4. Monge–Ampère system on T1H n+1 as K = c in Hn+1. In general, let us consider Rn+2 r = Rn+2 with the inner product x · y = − r−1∑ i=0 xiyi + n+1∑ j=r xjyj , of signature (r, n+ 2− r). We set, for ε1 = 0,±1, ε2 = 0,±1, and for a real number a, S2n+1 ε1,ε2,a = { (x, y) ∈ Rn+2 r ×Rn+2 r \|x · x = ε1, y · y = ε2, x · y = a, x 6= 0, y 6= 0 } , provided that S2n+1 ε1,ε2,a 6= ∅. Moreover we set Sn+1 ε = {x ∈ Rn+2 r \|x · x = ε, x 6= 0} for ε = 0,±1. On S2n+1 ε1,ε2,a, the contact structure D is defined by θ = − r−1∑ i=0 yidxi + n+1∑ j=r yjdxj . We have the double Legendrian fibration M = S2n+1 ε1,ε2 π1 ww π2 '' W1 = Sn+1 ε1 W2 = Sn+1 ε2 by π1(x, y) = x and π2(x, y) = y. In the case where r = 1, ε1 = −1, ε2 = 1, a = 0, since S2n+1 −1,1,0 = T1H n+1 = T−1S n+1 1 ⊂ Rn+2 1 ×Rn+2 1 , Sn+1 −1 = Hn+1 : the hyperbolic space, Sn+1 1 : the de Sitter space, we have the double Legendrian fibration M = T1H n+1 ∼= Hn+1 × Sn π1 tt π2 ** W1 = Hn+1 W2 = Sn+1 1 (cf. the hyperbolic Gauss map [15]). Lemma 8.12. The above bi-Legendrian fibration is contactomorphic to the standard model. The Lagrangian pair (E1, E2) = (Ker(π2)∗,Ker(π1)∗) is flat. Proof. For instance, on the open subset {x0 6= 0, yn+1 6= 0}, we take the system of coordinates x′i = xi x0 , p′i = yi yn+1 , z′ = −xn+1 x0 . Moreover we set z̃′ = y0 yn+1 . Then we have z′+ z̃′+ n∑ i=1 x′ip ′ i = 0, and θ = −x0yn+1 ( dz′ − n∑ i=1 p′idx ′ i ) . Then we see that the bi-Legendrian fibration is contacto- morphic to the standard model. � 30 G. Ishikawa and Y. Machida We set Ω1 = c iX(dx0 ∧ · · · ∧ dxn+1), the standard volume form on W1 = Hn+1 multiplied by c ∈ R (c 6= 0), and set Ω2 = iY (dy0 ∧ · · · ∧ dyn+1), the standard volume form on W2 = Sn+1 1 . Here X = −x0 ∂ ∂x0 + n+1∑ i=1 xi ∂ ∂xi and Y = −y0 ∂ ∂y0 + n+1∑ i=1 yi ∂ ∂yi . The Reeb vector field R on T1H n+1 is given by R = n+1∑ i=0 ( yi ∂ ∂xi + xi ∂ ∂yi ) . Then we have the following. Proposition 8.13. The associated Monge–Ampère system M with Lagrangian pair on M = T1H n+1 is generated by θ and ω = iR ( π∗1Ω1 − π∗2Ω2 ) = c n+1∑ i=1 (−1)ix0yidx1 ∧ · · · ∧ ˘dxi ∧ · · · ∧ dxn+1 + c ∑ 0≤j<i≤n+1 (−1)i+jxiyjdx0 ∧ · · · ∧ ˘dxj ∧ · · · ∧ ˘dxi ∧ · · · ∧ dxn+1 − c ∑ 1≤i<k≤n+1 (−1)i+kxiykdx0 ∧ · · · ∧ ˘dxi ∧ · · · ∧ ˘dxk ∧ · · · ∧ dxn+1 + n+1∑ i=1 (−1)iy0xidy1 ∧ · · · ∧ d̆yi ∧ · · · ∧ dyn+1 + ∑ 0≤j<i≤n+1 (−1)i+jyixjdy0 ∧ · · · ∧ ˘dyj ∧ · · · ∧ d̆yi ∧ · · · ∧ dyn+1 − ∑ 1≤i<k≤n+1 (−1)i+kyixkdy0 ∧ · · · ∧ d̆yi ∧ · · · ∧ ˘dyk ∧ · · · ∧ dyn+1. The Gaussian curvature of the projection to W1 = Hn+1 of a geometric solution of M is equal to constant c outside of singular locus. Remark 8.14. The group G = O(1, n + 1) acts on W1 = Hn+1, on W2 = Sn+1 1 and on Hn+1 × Sn+1 1 , thus on T1H n+1 = {(x, y) ∈ Hn+1 × Sn+1 1 \|x · y = 0}. The contact structure D = {θ = 0} and the contact form θ = −y0dx0 + n+1∑ i=1 yidxi are G-invariant, and any contact form defining D is a non-zero constant multiple of θ. We get the diagram: W1 = Hn+1 = G/H ′ ←M = T1H n+1 = G/H →W2 = Sn+1 1 = G/H ′′, for the isotropy groups H, H ′ and H ′′ satisfying H ′ ∼= O(n+ 1)←↩ H ∼= O(n) ↪→ H ′′ ∼= O(1, n). The G-invariant volume forms on W1 = Hn+1 and W2 = Sn+1 1 are unique up to non-zero constant. We can construct the Monge–Ampère system M = 〈θ, dθ, ω〉 with Lagrangian pair globally on M = T1H n+1. Monge–Ampère Systems with Lagrangian Pairs 31 8.5. The Monge–Ampère systems introduced in Sections 8.1–8.4 are all Euler–Lagrange Monge–Ampère systems. In fact, by Lemmas 8.4, 8.8, 8.12, there exists a system of local coordinates (x′, z′, p′) = (x′1, . . . , x ′ n, z ′, p′1, . . . , p ′ n) at any point of M such that a local contact form is given by θ = dz′ − n∑ i=1 p′idx ′ i, the Lagrangian pair is given by E1 = {v ∈ TM \| θ(v) = 0, dp′1(v) = 0, . . . , dp′n(v) = 0}, E2 = {u ∈ TM \| θ(u) = 0, dx′1(u) = 0, . . . , dx′n(u) = 0}, the Monge–Ampère system M is generated by an n-form ω̃ = f(x′1, . . . , x ′ n, z ′)dx′1 ∧ · · · ∧ dx′n − g ( p′1, . . . , p ′ n, n∑ i=1 x′ip ′ i − z′ ) dp′1 ∧ · · · ∧ dp′n for some (pulled-back) functions f , g on W1, W2 respectively. If a local contactomorphism Φ: M → R2n+1 and diffeomorphisms ϕ : W1 → Rn+1, ψ : W2 → Rn+1 give the contactomor- phism of the bi-Legendrian fibration to the standard model, then we have( ϕ−1 )∗ Ω1 = f(x, z)dz ∧ dx1 ∧ · · · ∧ dxn, ( ψ−1 )∗ Ω2 = g(p, z̃)dz̃ ∧ dp1 ∧ · · · ∧ dpn, for some non-zero functions f , g. For example, we calculate f , g explicitly in Euclidean geometry (Section 8.2), for the system of coordinates x′i = xi, p ′ i = −yi/yn+1 (1 ≤ i ≤ n), z′ = xn+1 on the open set {yn+1 6= 0}. A direct calculation yields that the Monge–Ampère system for K = c is generated by ω̃ = c dx′1 ∧ · · · ∧ dx′n − (−1)n ( 1 + p′ 2 1 + · · ·+ p′ 2 n )−n+2 2 dp′1 ∧ · · · ∧ dp′n, with the contact form θ = dz′ − n∑ i=1 p′idx ′ i. Since (ϕ−1)∗Ω1 and (ψ−1)∗Ω2 are local volume forms on Rn+1, the argument in Example 7.3 and Lemma 7.1 yields that all Monge–Ampère systems introduced in Sections 8.1–8.4 are Euler– Lagrange Monge–Ampère systems. Acknowledgements The first author was partially supported by Grants-in-Aid for Scientific Research No. 19654006. The second author was partially supported by Grants-in-Aid for Scientific Research (C) No. 18540105. The authors would like to thank anonymous referees for the valuable comments to improve the paper. 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Japan, Tokyo, 1993, 413–494. http://arxiv.org/abs/math.DG/0207039 http://dx.doi.org/10.1090/S0894-0347-01-00366-6 http://arxiv.org/abs/math.DG/0003099 http://dx.doi.org/10.1007/978-1-4613-9714-4 http://dx.doi.org/10.1007/978-0-8176-4771-1 http://dx.doi.org/10.1002/9781118032527 http://dx.doi.org/10.1142/S0129167X06003485 http://arxiv.org/abs/math.DG/0502154 http://dx.doi.org/10.1016/S0926-2245(01)00033-X http://dx.doi.org/10.1112/S0024611502013850 http://dx.doi.org/10.1007/978-3-642-61981-6 http://dx.doi.org/10.1007/978-3-642-61981-6 http://arxiv.org/abs/1303.1307 http://dx.doi.org/10.1070/RM1979v034n01ABEH002873 http://dx.doi.org/10.1016/0926-2245(93)90004-K http://dx.doi.org/10.2969/jmsj/01920215 http://dx.doi.org/10.14492/hokmj/1381758416 1 Introduction 2 Monge-Ampère systems and Lagrangian pairs 3 Lagrangian pair and bi-decomposable form 4 Lagrangian contact structures 5 Automorphisms of Monge-Ampère systems with Lagrangian pairs 6 Hesse representations 7 A method to construct Monge-Ampère systems with Lagrangian pairs 8 Homogeneous Monge–Ampère systems with Lagrangian pairs References
id	nasplib_isofts_kiev_ua-123456789-147154
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn	1815-0659
language	English
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publishDate	2015
publisher	Інститут математики НАН України
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spelling	Ishikawa, G. Machida, Y. 2019-02-13T17:48:28Z 2019-02-13T17:48:28Z 2015 Monge-Ampère Systems with Lagrangian Pairs / G. Ishikawa, Y. Machida // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 31 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 58K20; 53A15; 53C42 DOI:10.3842/SIGMA.2015.081 https://nasplib.isofts.kiev.ua/handle/123456789/147154 The classes of Monge-Ampère systems, decomposable and bi-decomposable Monge-Ampère systems, including equations for improper affine spheres and hypersurfaces of constant Gauss-Kronecker curvature are introduced. They are studied by the clear geometric setting of Lagrangian contact structures, based on the existence of Lagrangian pairs in contact structures. We show that the Lagrangian pair is uniquely determined by such a bi-decomposable system up to the order, if the number of independent variables ≥3. We remark that, in the case of three variables, each bi-decomposable system is generated by a non-degenerate three-form in the sense of Hitchin. It is shown that several classes of homogeneous Monge-Ampère systems with Lagrangian pairs arise naturally in various geometries. Moreover we establish the upper bounds on the symmetry dimensions of decomposable and bi-decomposable Monge-Ampère systems respectively in terms of the geometric structure and we show that these estimates are sharp (Proposition 4.2 and Theorem 5.3). The first author was partially supported by Grants-in-Aid for Scientific Research No. 19654006. The second author was partially supported by Grants-in-Aid for Scientific Research (C) No. 18540105. The authors would like to thank anonymous referees for the valuable comments to improve the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Monge-Ampère Systems with Lagrangian Pairs Article published earlier
spellingShingle	Monge-Ampère Systems with Lagrangian Pairs Ishikawa, G. Machida, Y.
title	Monge-Ampère Systems with Lagrangian Pairs
title_full	Monge-Ampère Systems with Lagrangian Pairs
title_fullStr	Monge-Ampère Systems with Lagrangian Pairs
title_full_unstemmed	Monge-Ampère Systems with Lagrangian Pairs
title_short	Monge-Ampère Systems with Lagrangian Pairs
title_sort	monge-ampère systems with lagrangian pairs
url	https://nasplib.isofts.kiev.ua/handle/123456789/147154
work_keys_str_mv	AT ishikawag mongeamperesystemswithlagrangianpairs AT machiday mongeamperesystemswithlagrangianpairs

Monge-Ampère Systems with Lagrangian Pairs

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