Invariant Nijenhuis Tensors and Integrable Geodesic Flows

We study invariant Nijenhuis (1,1)-tensors on a homogeneous space G/K of a reductive Lie group G from the point of view of integrability of a Hamiltonian system of differential equations with the G-invariant Hamiltonian function on the cotangent bundle T*(G/K). Such a tensor induces an invariant Poi...

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Published in:	Symmetry, Integrability and Geometry: Methods and Applications
Date:	2019
Main Authors:	Lompert, K., Panasyuk, A.
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Language:	English
Published:	Інститут математики НАН України 2019
Online Access:	https://nasplib.isofts.kiev.ua/handle/123456789/210239
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Cite this:	Invariant Nijenhuis Tensors and Integrable Geodesic Flows / K. Lompert, A. Panasyuk // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 33 назв. — англ.

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author	Lompert, K. Panasyuk, A.
author_facet	Lompert, K. Panasyuk, A.
citation_txt	Invariant Nijenhuis Tensors and Integrable Geodesic Flows / K. Lompert, A. Panasyuk // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 33 назв. — англ.
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container_title	Symmetry, Integrability and Geometry: Methods and Applications
description	We study invariant Nijenhuis (1,1)-tensors on a homogeneous space G/K of a reductive Lie group G from the point of view of integrability of a Hamiltonian system of differential equations with the G-invariant Hamiltonian function on the cotangent bundle T(G/K). Such a tensor induces an invariant Poisson tensor Π₁ on T(G/K), which is Poisson compatible with the canonical Poisson tensor ΠT(G/K). This Poisson pair can be reduced to the space of G-invariant functions on T(G/K) and produces a family of Poisson commuting G-invariant functions. We give, in Lie algebraic terms, necessary and sufficient conditions for the completeness of this family. As an application we prove Liouville integrability in the class of analytic integrals polynomial in momenta of the geodesic flow on two series of homogeneous spaces G/K of compact Lie groups G for two kinds of metrics: the normal metric and new classes of metrics related to decomposition of G to two subgroups G=G₁⋅G₂, where G/Gᵢ are symmetric spaces, K=G₁∩G₂.
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 15 (2019), 056, 30 pages Invariant Nijenhuis Tensors and Integrable Geodesic Flows Konrad LOMPERT † and Andriy PANASYUK ‡ † Faculty of Mathematics and Information Science, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warszawa, Poland E-mail: k.lompert@mini.pw.edu.pl ‡ Faculty of Mathematics and Computer Science, University of Warmia and Mazury, ul. S loneczna 54, 10-710 Olsztyn, Poland E-mail: panas@matman.uwm.edu.pl Received December 19, 2018, in final form August 02, 2019; Published online August 07, 2019 https://doi.org/10.3842/SIGMA.2019.056 Abstract. We study invariant Nijenhuis (1, 1)-tensors on a homogeneous space G/K of a reductive Lie group G from the point of view of integrability of a Hamiltonian system of differential equations with the G-invariant Hamiltonian function on the cotangent bun- dle T ∗(G/K). Such a tensor induces an invariant Poisson tensor Π1 on T ∗(G/K), which is Poisson compatible with the canonical Poisson tensor ΠT∗(G/K). This Poisson pair can be reduced to the space of G-invariant functions on T ∗(G/K) and produces a family of Poisson commuting G-invariant functions. We give, in Lie algebraic terms, necessary and sufficient conditions of the completeness of this family. As an application we prove Liouville integra- bility in the class of analytic integrals polynomial in momenta of the geodesic flow on two series of homogeneous spaces G/K of compact Lie groups G for two kinds of metrics: the normal metric and new classes of metrics related to decomposition of G to two subgroups G = G1 ·G2, where G/Gi are symmetric spaces, K = G1 ∩G2. Key words: bi-Hamiltonian structures; integrable systems; homogeneous spaces; Lie alge- bras; Liouville integrability 2010 Mathematics Subject Classification: 37J15; 37J35; 53D25 1 Introduction By Maupertuis’s principle integrability of the geodesic flow of a (pseudo-)Riemannian metric is a question as old as classical mechanics itself. In this paper we consider Hamiltonian systems and understand integrability in the sense of Arnold–Liouville, i.e., as existence of a complete family of first integrals in involution. The Clairaut theorem on existence of linear integral for the motion of a free particle on a surface of revolution is traditionally mentioned as one of the first results on Arnold–Liouville integrability of geodesic flows. Next classical cases are the Euler top and geodesics on ellipsoid. In modern mathematical literature one could find many examples of integrable geodesic flows on homogeneous spaces of Lie groups starting probably with the papers [15, 30], see also the review [6] and references therein and later works [7, 12, 16, 17]. The present paper continues this line and develops a new approach for constructing integrable geodesic flows on homogeneous spaces. Let G be a reductive Lie group, K ⊂ G its closed subgroup. The cotangent bundle T ∗(G/K) with its canonical Poisson structure Π is a phase space of a Hamiltonian system with the Hamiltonian function equal to the quadratic form q of an G-invariant pseudo-Riemannian metric, which can be constructed as follows. Let 〈 , 〉 be an AdG-invariant symmetric bilinear form on g, the Lie algebra of G. It gives rise to a bi-invariant metric on G, which induces on G/K an G-invariant metric 〈 , 〉G/K called normal. Besides, one mailto:k.lompert@mini.pw.edu.pl mailto:panas@matman.uwm.edu.pl https://doi.org/10.3842/SIGMA.2019.056 2 K. Lompert and A. Panasyuk can consider a symmetric ad k-invariant linear operator (called inertia operator) nk⊥ : k⊥ → k⊥, where k is the Lie algebra of K and k⊥ is its orthogonal complement in g with respect to 〈 , 〉. It will give rise to an G-invariant (1, 1)-tensor N : T (G/K) → T (G/K), which is symmetric with respect to 〈 , 〉G/K , and to another G-invariant metric 〈·, ·〉N := 〈N ·, ·〉G/K . The question of integrability of the geodesic flows of both metrics 〈 , 〉G/K and 〈·, ·〉N on G/K consists of finding a family of dim(G/K) − 1 independent analytic and polynomial in momenta functions on T ∗(G/K) which Poisson commute with the quadratic form q and with each other. It is known [6, Section 5] that there are two families F1 and F2 of analytic functions on T ∗(G/K) that Poisson commute with each other, in which one can look for desirable integrals. These are the family F2 of G-invariant functions and the family F1 of functions of the form µ∗canf , where µcan : T ∗(G/K) → g∗ is the momentum map corresponding to the natural Hamiltonian action of G on T ∗(G/K) and f is an analytic function on g∗. Obviously q ∈ F2 and taking a family F of commuting polynomials on g∗ (by the Sadetov theorem [29] there exist complete such families, see also [3]) one gets the family A := µ∗can(F) of integrals of q polynomial in momenta. Thus the problem now is reduced to the following one: construct a family B ⊂ F2 of commuting polynomial in momenta integrals of q such that the family A+ B is complete. An approach for constructing such a family B was proposed in [17]. The homogeneous spaces considered were the coadjoint orbits O of G. A second G-invariant Poisson structure Π1 was constructed on T ∗(G/K) which is compatible with Π and the family B was the canonical family of functions in involution related with the Poisson pair (Π′,Π′1) being the reduction of the Poisson pair (Π,Π1) with respect to the action of G. Essential role in the construction of Π1 played the Kirillov–Kostant–Suriau symplectic form ωO on O, as Π1 = (ω + π∗ωO)−1, where ω = −Π−1 is the canonical symplectic form on T ∗O and π : T ∗O → O is the canonical projection. In this paper we propose a novel approach for constructing the family B. Similarly to the case above, we construct a second Poisson structure Π1 compatible with Π, but we use invariant Nijenhuis (1, 1)-tensors N : T (G/K)→ T (G/K) for this purpose instead, in particular avoiding the restriction on G/K of being a coadjoint orbit. In more detail, Π1 = Ñ ◦ Π, where Ñ is the so-called cotangent lift of N , see Definition 4.5. Obviously, an invariant (1, 1)-tensor on G/K is determined by a linear operator n : g→ g. We get some Lie algebraic conditions on this operator which are necessary and sufficient for the so-called kroneckerity of the Poisson pair (Π′,Π′1) obtained as the reduction of the pair (Π,Π1) and, as a consequence, of the completeness of the family B (and A + B), see Theorem 5.1, the main result of this paper, and Theorem 5.4. As an application we construct two series of invariant Nijenhuis (1, 1)-tensors on homogeneous spaces Gk/Kk of compact simple Lie groups, where (Gk,Kk) is (SU(2k), S(U(2k − 1)×U(1)) ∩ Sp(k)) or (SO(2k+2),SO(2k+1)∩U(k+1)), which lead to invariant metrics with geodesic flow Liouville integrable in the class of integrals analytic and polynomial in momenta (Theorem 6.2). Besides we prove integrability of the normal metric on these homogeneous spaces. Below the content of the paper is discussed in more detail. In Section 2 we study Lie algebraic conditions on the operator n : g→ g which guarantee the vanishing of the Nijenhuis torsion of N (Theorem 2.7) and consider some examples. A crucial role in our considerations play bi-Hamiltonian (bi-Poisson) structures, i.e., pencils of Poisson structures generated by pairs of compatible ones. We devote Section 3 to related notions and preparatory results which will enable us to study the completeness of families of functions in involution. Theorem 3.7 gives some criteria of completeness of the canonical family of G-invariant functions related to an action of a Lie group G on a bi-Poisson manifold M being Hamiltonian with respect to almost all Poisson structures from the pencil. The theorem requires some assumptions among which the most significant one says that the action of G on M is locally free. This assumption enables to use the so-called bifurcation lemma and to prove the constancy of rank of the reduced bi-Poisson structure for almost all values of the parameter, which is a first step for achieving the kroneckerity. Invariant Nijenhuis Tensors and Integrable Geodesic Flows 3 In Section 4 we study bi-Poisson structures on T ∗(G/K) generated by Poisson pairs (Π,Π1 = Ñ ◦ Π), where N is a semisimple invariant Nijenhuis (1, 1)-tensor. We show that almost all (generic) Poisson structures from the corresponding Poisson pencil are nondegenerate and cal- culate the dimensions of the symplectic leaves of the exceptional (not being generic) Poisson structures (Lemma 4.8). We prove the hamiltonicity of canonical action of G on T ∗(G/K) with respect to the generic Poisson structures, as well as the hamiltonicity of the actions of some subgroups (stabilizers of the symplectic leaves) on the symplectic leaves of the exceptional ones. We calculate the corresponding momentum maps (see Lemma 4.9) as well as these stabilizers (Lemma 4.10). The main result, Theorem 5.1, which gives necessary and sufficient conditions for kroneckerity of the reduced Poisson pair (Π′,Π′1) in terms of the indices of the Lie algebra g and some its contractions (see formula (5.3)), is proved in Section 5. As a corollary we prove Theorem 5.4 stating the complete integrability of the geodesic flow of the normal metric and the metric with the inertia operator n\|k⊥ under the assumption that the sufficient conditions from Theorem 5.1 are satisfied. In Section 6 we apply the above results to construct examples of metrics with integrable geodesic flow. The main idea which enables to fit conditions of Theorem 5.1 is based on the Brailov theorem (see Theorem 6.1) stating equality of indices of a semisimple Lie algebra and its Z2-contractions. We observe that among the examples of invariant Nijenhuis (1, 1)-tensors on a homogeneous space G/K from Section 2 related to the Onishchik list of decompositions g = g1+g2 of a simple compact Lie algebra to two subalgebras (Example 2.12) there are two series (g(k), g1(k), g2(k)) in which both the pairs (g(k), g1(k)) and (g(k), g2(k)) are symmetric, i.e., by the Brailov theorem these examples satisfy conditions (5.3) of Theorem 5.1 (the Lie algebra k of the group K is equal g1(k)∩ g2(k)). In order to apply this theorem for the proof of complete integrability of the geodesic flow one needs only ensure that the action of G on T ∗(G/K) is locally free. This is done in the proof of Theorem 6.2 stating the complete integrability of the geodesic flows of the normal metric and the metric with the corresponding inertia operator. The explicit formulae for the realizations of Lie algebras g(k), g1(k), g2(k) for both series as well as for the corresponding inertia operators are given in Appendix A. There we also indicate conditions under which these operators (and the corresponding metrics) are positive definite. We end the paper by concluding remarks (Section 7) in which we discuss some details of the paper and possible perspectives. Fix some notations. We write P : G → G/K, π : T ∗M → M , and p : M → M/G for the canonical projections. All objects in this paper are real analytic or complex analytic. Given a vector bundle E, we write Γ(E) for the space of sections of E, and E(M) will stand for the space of functions on a manifold M (of the corresponding category). 2 Invariant Nijenhuis tensors on homogeneous spaces Definition 2.1. Let M be a connected manifold. A (1, 1)-tensor field N : TM → TM is a Nijenhuis tensor if its Nijenhuis torsion vanishes, i.e., for any vector fields X,Y ∈ Γ(TM): TN (X,Y ) := [NX,NY ]−N [X,Y ]N = 0, where we put [X,Y ]N := [NX,Y ] + [X,NY ]−N [X,Y ]. Similarly, given any Lie algebra (g, [ , ]), a linear operator n : g → g is an algebraic Nijenhuis operator if it satisfies Tn(X,Y ) := [nX, nY ]−n([nX, Y ] + [X,nY ]−n[X,Y ]) = 0 for all vectors X,Y ∈ g (cf. [9, 13]). 4 K. Lompert and A. Panasyuk Let an action of a Lie group G on a manifold M be given. Definition 2.2. We say that a (1, 1)-tensor field N : TM → TM is G-invariant if for any element of the Lie group g ∈ G, the tensor N commutes with the tangent map g∗ : TM → TM to the diffeomorphism g : M →M , i.e., the following diagram is commutative TM N−→ TM ↓ g∗ ↓ g∗ TM N−→ TM. A distribution of subspaces Dx ⊂ TxM is G-invariant, if for any g ∈ G and any x ∈M we have g∗,x(Dx) = Dgx. The following lemmas are crucial ingredients in further considerations. Let G be any Lie group and K a closed Lie subgroup of G (the quotient M = G/K is then a smooth G-manifold). Lemma 2.3. Let N : T (G/K) → T (G/K) be a semisimple (1, 1)-tensor and assume that N is G-invariant. Then the eigenvalues of N are constant. Proof. Since the operator N is G-invariant, it follows that its eigenfunctions are also G- invariant, therefore on homogeneous space they are constant functions. � Given a real manifold M , we write TCM for the complexified tangent bundle to M . Lemma 2.4. Let G/K be a real homogeneous space. There is a one-to-one correspondence between G-invariant distributions D ⊂ TC(G/K) and subspaces d ⊂ gC such that kC ⊂ d and[ kC, d ] ⊂ d (here g, k ⊂ g are the Lie algebras of the Lie groups G,K ⊂ G). An G-invariant distribution D is involutive if and only if the subspace d is a subalgebra in gC. Moreover, D is real, i.e., D = D, where the bar stands for the complex conjugation on TC(G/K), if and only if so is d, i.e., d = d̄, where the bar denotes the complex conjugation in gC with respect to the real form g. Proof. Below we let P : G → G/K to denote the canonical projection. An invariant distribu- tion D on G/K defines the distribution D̂ := P−1 ∗ (D) ⊂ TCG, which by construction is left G-invariant. Indeed, the invariance of D, g∗,x(Dx) = Dgx, implies Lg-invariance of D̂ as the commutativity of the following diagram shows TyG Lg,∗\|y−→ TgyG ↓ P∗,y ↓ P∗,gy Tx(G/K) g∗,x−→ Tgx(G/K); here Lg is the left translation by g and y ∈ G is so that P (y) = x. Moreover, D̂ is right K-invariant. To show this observe that, since P is a surjective sub- mersion, in a vicinity of points g ∈ G and P (g) ∈ G/K there exist local coordinate sys- tems (x1, . . . , xm, y1, . . . , yk) and (x′1, . . . , x ′ m) respectively such that P (x1, . . . , xm, y1, . . . , yk) = (x′1, . . . , x ′ m), x′i = xi, i = 1, . . . ,m. Let X1(x′), . . . , Xl(x ′), Xr(x ′) = Xi r(x ′ 1, . . . , x ′ m) ∂ ∂x′i , be local linearly independent vector fields on G/K generating the distribution D. Then the distribu- tion D̂ is generated by the vector fields X̂r(x) = Xi r(x1, . . . , xm) ∂ ∂xi , r = 1, . . . , l, and Y1, . . . , Yk, where the last ones are the fundamental vector fields of the right K-action. These last are tan- gent to the fibers of P , locally can be expressed as combinations of ∂ ∂yj and vice versa, ∂ ∂yj can Invariant Nijenhuis Tensors and Integrable Geodesic Flows 5 be locally expressed as combinations of Y1, . . . , Yk. Obviously, [ Yi, X̂j ] = f sijYs for some func- tions f sij , which together with the involutivity of the system of vector fields {Y1, . . . , Yk} gives[ Yi, D̂ ] ⊂ D̂. Let d := D̂e ⊂ TC e G, where e ∈ G is the neutral element. The left and right K-invariance of D̂ implies, under the identification TC e G ∼= gC, the Ad(K)-invariance of the subspace d ⊂ gC, or, on the infinitesimal level, its ad(k)-invariance: [ kC, d ] ⊂ d. Now if D is involutive, then so is D̂. Indeed, the systems of vector fields {Xj} and, conse- quently, { X̂j } are involutive. Hence so is the total system of vector fields { Yi, X̂j } . Infinitesi- mally this can be expressed as [d, d] ⊂ d. Vice versa, let d ⊂ gC ∼= TC e G be an ad k-invariant subspace. Define a distribution D̂ ⊂ TCG by D̂g = Lg,∗d. Then D̂ is left G-invariant and right K-invariant and descends to a uniquely defined invariant distribution D ⊂ TC(G/K) by means of the complexified tangent map PC ∗ : TCG→ TC(G/K). If d ⊂ gC is a subalgebra, then clearly the distribution D̂ is involutive. Moreover, from the above local description it follows that the system of vector fields { X̂j } is involutive and P∗X̂j = Xj , and, as a consequence, so is the system {Xj}. ThereforeD ⊂ TC(G/K) is involutive. The last assertion of the lemma is obvious. � Lemma 2.5. Let D ⊂ T (G/K) be an G-invariant integrable distribution on G/K relative to a subalgebra h ⊂ g, h ⊃ k (as in Lemma 2.4 but we admit also the complex analytic case), and let H ⊂ G be the corresponding subgroup. Denote by P : G → G/K the canonical projection. Then 1) the leaves of the foliation tangent to D are the projections with respect to P of the left cosets gH, g ∈ G; 2) given ξ ∈ g, the fundamental vector field Xξ of the G-action on G/K is tangent to the leaf P (gH) if and only if ξ ∈ Adg h ⊂ g. Proof. Consider the integrable distribution D̂ built in the proof of Lemma 2.4. Then it is easy to see that the foliation tangent to D̂ coincides with the foliation of the left cosets gH, g ∈ G. Since P∗ ( D̂ ) = D, the leaves of the corresponding foliations are projected on each other by means of P , which proves item 1. The right invariant vector field ξR on G, ξR\|e = ξ, is tangent to gH at the point gh ∈ gH if and only if ξR(gh) ∈ Tgh(gH) ⇔ Rgh,∗(ξ) ∈ Tgh(gH) = Lgh,∗h ⇔ ξ ∈ R(gh)−1,∗Lgh,∗h = Adgh h = Adg h (here ξ ∈ g ∼= TeG). Hence Xξ = P∗ξR is tangent to P (gH) if and only if ξ ∈ Adg h. � Lemma 2.6. Let N : TM → TM be a semisimple (1, 1)-tensor with constant distinct eigenvalues λ1, . . . , λs ∈ C (or λ1, . . . , λs ∈ R) and let Di ⊂ TCM (or, respectively Di ⊂ TM) be the eigendistribution corresponding to λi. Then TN = 0 if and only if the distributions Di and Di +Dj are involutive for any i, j. Proof. Assume N is Nijenhuis. It is easy to see that TN−λI = TN = 0 for any λ ∈ C. In particular [( NC − λiI ) X, ( NC − λiI ) Y ] = ( NC − λiI ) [X,Y ]NC−λiI for any vector fields X, Y and the image of NC − λiI : TCM → TCM is an integrable distribution. As a consequence, Di = ⋂ k 6=i im ( NC − λkI ) and Di +Dj = ⋂ k 6=i,j im ( NC − λkI ) are integrable. Now, let the decomposition TCM = D1 ⊕ · · · ⊕Ds be such that Di + Dj are integrable for any i, j. By the bilinearity of Nijenhuis torsion tensor it is enough to prove that TN (x, y) = 0 for x ∈ Γ(Di), y ∈ Γ(Dj), 1 ≤ i, j ≤ n: TN (x, y) = [Nx,Ny]−N([Nx, y] + [x,Ny]) +N2[x, y] 6 K. Lompert and A. Panasyuk = λiλj([x, y]i + [x, y]j)−N(λi([x, y]i + [x, y]j) + λj([x, y]i + [x, y]j)) +N(λi[x, y]i + λj [x, y]j) = λiλj([x, y]i + [x, y]j)− ( λ2 i [x, y]i + λiλj [x, y]j + λiλj [x, y]i + λ2 j [x, y]j ) + (λ2 i [x, y]i + λ2 j [x, y]j) = 0 (here we denote by [x, y]i the i-th component of the element [x, y] with respect to the decompo- sition above). The proof in the case of real eigenvalues is analogous. � Let G be any Lie group and g = Lie(G) its Lie algebra, K a closed Lie subgroup of G and k = Lie(K). Theorem 2.7. There is a one-to-one correspondence between (i) G−invariant semisimple Nijenhuis (1, 1)-tensors N : T (G/K) → T (G/K) with the spec- trum {λ1, . . . , λs}, where λi are distinct, λ1, . . . , λ2p ∈ C, λi = λi+p for i = 1, . . . , p and λ2p+1, . . . , λs ∈ R, and (ii) decompositions gC = g1 + · · ·+ gs of gC to the sum of subspaces such that: 1) ∀i,j∈{1,...,s},i 6=j gi ∩ gj = kC; 2) the induced decomposition of the factor space gC/kC is direct: gC/kC = ( g1/k C)⊕· · ·⊕( gs/k C); 3) ∀i,j∈{1,...,s} gi + gj are Lie subalgebras in gC; 4) gi = gi+p for i = 1, . . . , p and gj = gj for j = 2p+ 1, . . . , s. The decomposition (ii) induces the decomposition TC(G/K) = D1 ⊕ · · · ⊕ Ds to involutive subbundles, the corresponding (1, 1)-tensor N is then given by N \|Di = λi IdDi and, vice versa, given N as in (i) one constructs the decomposition (ii) by the decomposition TC(G/K) = D1 ⊕ · · · ⊕Ds of TC(G/K) to the eigendistributions of N . Proof. Let N be an G-invariant semisimple Nijenhuis (1, 1)-tensor on G/K with the spectrum {λ1, . . . , λs; λi ∈ C, λi 6= λj , for i 6= j}. From Lemma 2.6 it follows that there is a decompo- sition TC(G/K) = D1 ⊕ · · · ⊕Ds into integrable distributions, which, as the eigenspaces of an G-invariant tensor, are also G-invariant. By Lemma 2.4 there is a one-to-one correspondence between G-invariant distributions Di and subalgebras gi containing kC, hence there is a decom- position of gC = g1 + · · ·+ gs, such that gi ∩ gj = kC for any i 6= j. Applying Lemma 2.4 to the sum of distributions Di +Dj we see that it is involutive if and only if gi + gj is a subalgebra. Item 3 follows from the last assertion of Lemma 2.4 and from the obvious fact that Di = Di+p for i = 1, . . . , p and Dj = Dj for j = 2p+ 1, . . . , s. The proof in reverse direction follows the same argumentation with the use of the equivalences in lemmas cited. � Below we present some examples for which decompositions of Lie algebras mentioned in Theorem 2.7 are given explicitly. First series of examples come from semisimple algebraic Nijenhuis operators n : g→ g, which are ad k-invariant for some Lie subalgebra k ⊂ g, i.e., n ◦ ad k = ad k ◦ n for all k ∈ k. Then by ad k-invariance we can extend it to an invariant Nijenhuis (1, 1)-tensor N on G/K. In the literature the following two classes of algebraic Nijenhuis operators are widely known [9, 13, 27]:1 first is related to a direct decomposition of the algebra g to two subalgebras, second 1The is one more class defined on the full matrix algebra by nX = AXB + BAX, where A2 = B2 = I [19]. For some particular cases of the matrices A and B the corresponding operator is semisimple. However these cases are beyond the scope of this paper. Invariant Nijenhuis Tensors and Integrable Geodesic Flows 7 is related to the operator of left multiplication on the full matrix algebra. Below we consider particular cases of these two classes. Example 2.8. Let g be a semisimple Lie algebra with the root system R with respect to a Cartan subalgebra h ⊂ g. Let g = h + ∑ α∈R gα be the corresponding root decomposition. Choose R+ and R− to be sets of positive and negative roots and let S ⊂ Π be any subset of the set of positive simple roots. We denote by [S] the set of positive roots generated by S. Consider the decomposition g = p⊕ p⊥, where p := h + ∑ α∈R− gα + ∑ α∈[S] gα is the corresponding parabolic subalgebra and p⊥ = ∑ α∈R+\[S] gα (the orthogonal complement with respect to Killing form). Then p⊥ is obviously a subalgebra too. The operator n : g→ g defined by n\|g1 = λ1 Idg1 , n\|g2 = λ2 Id \|g2 with g1 = p and g2 = p⊥ and with arbitrary λ1, λ2 is algebraic Nijenhuis (cf. [9, 24]). One may take k = p∩popposite, where popposite = h+ ∑ α∈−[S]⊂R− gα+ ∑ α∈R+ gα. Then the operator n will be ad k-invariant and will generate an G-invariant Nijenhuis (1, 1)-tensor on G/K, where G, K ⊂ G are the corresponding Lie groups. The decomposition of Theorem 2.7 looks as follows: g1 := p, g2 := popposite. An instance of such a situation for g = sl(3,R) can be schematically presented as p = ∗ 0 0 ∗ ∗ ∗ ∗ ∗ ∗  , p⊥ = 0 ∗ ∗ 0 0 0 0 0 0  , popposite = ∗ ∗ ∗0 ∗ ∗ 0 ∗ ∗  , k = ∗ 0 0 0 ∗ ∗ 0 ∗ ∗  , where the corresponding set S consists of the sole root e2 − e3, ei(H) being the i-th diagonal element of H ∈ h. Example 2.9. Let g = gl(n,K), K = R,C, and consider n = LA, the operator of left multi- plication by a matrix A ∈ g. Then it is easy to see that n is an algebraic Nijenhuis operator. Taking A = diag(λ1, . . . , λn), λi 6= λj , i 6= j, we get a semisimple operator, whose eigenspaces ker(n− λi Id) consist of matrices with the only nonzero i-th row. Obviously, n is ad k-invariant for k = Z(A), the centralizer of A, which coincides with the subalgebra of diagonal matrices. The decomposition of Theorem 2.7 is g = n∑ i=1 gi, where gi = ker(n − λi Id) + k consists of the matrices having non zero elements at most on the diagonal and i-th row. The generalization to the case when multiplicities in the spectrum of A are admitted is straightforward. This example has also an obvious generalization to the case g = sl(n,K). Our next example is quite classical, as this is the complex structure operator on the adjoint orbits of the compact Lie groups which was intensively studied in the literature. We adapt the description of this operator to our notations. An alternative description can be found in [1, Chapter 8.B]. Example 2.10. Let g be a complex semisimple Lie algebra, h ⊂ g a Cartan subalgebra, g = h+∑ α∈R gα the corresponding root grading. For any α ∈ R choose Eα ∈ gα such that 〈Eα, E−α〉 = 1 and putHα := [Eα, E−α]. Then u = ∑ α∈R+ R(iHα)+ ∑ α∈R+ R(Eα−E−α)+ ∑ α∈R+ R(i(Eα+E−α)) ⊂ g, where R+ ⊂ R is a subset of positive roots, is the compact real form of g [11, Theorem 6.3, Chapter III]. By [10, Theorem 1.3, Chapter 6] the centralizer Zu(a) of any element a ∈ u (which is necessarily semisimple) is of the form Zu(a) = ∑ α∈R+ R(iHα)+ ∑ α∈[S] R(Eα−E−α)+ ∑ α∈[S] R(i(Eα+ E−α)), where S ⊂ R+ is a subset of the set of simple positive roots (cf. Example 2.8). Consider 8 K. Lompert and A. Panasyuk the operator j : u⊥ → u⊥, where u⊥ := ∑ α∈R+\[S] R(Eα − E−α) + ∑ α∈R+\[S] R(i(Eα + E−α)), given by j(Eα − E−α) = i(Eα + E−α), j(i(Eα + E−α)) = −(Eα − E−α). Note that jC(Eα) = iEα, jC(E−α) = −iE−α. The eigenspaces g′1 := ∑ α∈R+\[S] C(Eα) and g′2 := ∑ α∈R+\[S] C(E−α) are subalgebras as well as the subspaces gi := g′i ⊕ kC, k = Zu(a). Hence by Theorem 2.7 the operator j induces an invariant integrable almost complex structure on U/K, where U,K ⊂ U are the Lie groups corresponding to the Lie algebras u, k. We conclude that, although this operator is not arising from an algebraic Nijenhuis operator, the corresponding decomposition in fact coincides with that from Example 2.8). Now we come to a series of examples of different nature.2 The decomposition of Theorem 2.7 will still consist of two components which now need not be symmetric with respect to the involution interchanging gα and g−α. In other words, any decomposition g = g1 + g2 of a Lie algebra g to two subalgebras can be taken into consideration (with k = g1 ∩ g2). One of possible natural generalizations of Example 2.8 is considering two “nonsymmetric” parabolic subalgebras. Their intersection is the so-called seaweed subalgebra. Example 2.11. Let g be a semisimple Lie algebra with the root system R with respect to a Cartan subalgebra h ⊂ g. Let g = h + ∑ α∈R gα be the corresponding root decomposition. Choose R+ and R− to be sets of positive and negative roots and let S, S′ ⊂ Π be any subsets of the set of positive simple roots. Consider the parabolic subalgebras g1 = h + ∑ α∈R− gα + ∑ α∈[S] gα and g2 = h + ∑ α∈R+ gα + ∑ α∈−[S′] gα. An instance of such a situation for g = sl(3,R) can be schematically presented as g1 = ∗ 0 0 ∗ ∗ ∗ ∗ ∗ ∗  , g2 = ∗ ∗ ∗∗ ∗ ∗ 0 0 ∗  , k = ∗ 0 0 ∗ ∗ ∗ 0 0 ∗  , where the corresponding sets S and S′ consist of the roots e2 − e3 and e1 − e2 respectively, cf. Example 2.8. In [20] A.L. Onishchik classified all decompositions g = g1 + g2 for compact simple Lie algebras g and we list them below. (In [21] he also gave a classification of decompositions of reductive Lie algebras g to two subalgebras reductive in g, but we omit this case here.) Example 2.12. Let g be a compact simple Lie algebra. The following table presents all pairs of subalgebras (g1, g2) such that g = g1 + g2 together with possible embeddings i′ : g1 → g, i′′ : g2 → g up to conjugations. Below N stands for the trivial representation, ϕi for the specific representation mentioned in [20] and T for the 1-dimensional Lie algebra. 3 Bi-Poisson structures, kroneckerity, G-invariance, and complete families of functions in involution If M is a real or complex analytic manifold, E(M) will stand for the space of analytic functions on M in the corresponding category. We will write K for the corresponding ground field. We 2By this we mean that they are not necessarily related with an ad k-invariant algebraic Nijenhuis operator on the Lie algebra g. For instance, in Example 2.11 there are two ways to build a compatible with the decomposition g = g1 + g2 algebraic Nijenhuis operator n : g→ g: n\|g1 = λ1 Idg1 , n\|g′2 = λ2 Idg′2 with g′2 = g⊥1 , or n\|g2 = β2 Idg2 , n\|g′1 = β1 Idg′1 with g′1 = g⊥2 . However, in both the cases the operator in general will not be ad k-invariant. Concerning the decompositions from the Onishchik list, see Example 2.12, it seems that it is even impossible to build a compatible Nijenhuis operator for some of them. Invariant Nijenhuis Tensors and Integrable Geodesic Flows 9 g g1 i′ g2 i′′ k = g1 ∩ g2 restrictions A2n−1 Cn ϕ1 A2n−2 ϕ1 +N Cn−1 n > 1 A2n−1 Cn ϕ1 A2n−2 ⊕ T ϕ1 +N Cn−1 ⊕ T n > 1 B3 G2 ϕ2 B2 ϕ1 + 2N A1 B3 G2 ϕ2 B2 ⊕ T ϕ1 + 2N A1 ⊕ T B3 G2 ϕ2 D3 ϕ1 +N A2 Dn+1 Bn ϕ1 +N An ϕ1 + ϕn An−1 n > 2 Dn+1 Bn ϕ1 +N An ⊕ T ϕ1 + ϕn An−1 ⊕ T n > 2 D2n B2n−1 ϕ1 +N Cn ϕ1 + ϕ1 Cn−1 n > 1 D2n B2n−1 ϕ1 +N Cn ⊕ T ϕ1 + ϕ1 Cn−1 ⊕ T n > 1 D2n B2n−1 ϕ1 +N Cn ⊕A1 ϕ1 + ϕ1 Cn−1 ⊕A1 n > 1 D8 B7 ϕ1 +N B4 ϕ4 B3 D4 B3 ϕ3 B2 ϕ1 + 3N A1 D4 B3 ϕ3 B2 ⊕ T ϕ1 + 3N A1 ⊕ T D4 B3 ϕ3 B2 ⊕A1 ϕ1 + 3N A1 ⊕A1 D4 B3 ϕ3 D3 ϕ1 + 2N A2 D4 B3 ϕ3 D3 ⊕ T ϕ1 + 2N A2 ⊕ T D4 B3 ϕ3 B3 ϕ1 +N G2 recall basic definitions and concepts related to bi-Poisson structures, their kroneckerity and invariance (cf. [17]). We will say that some functions from the set E(M) are independent at a point x ∈M if their differentials are independent at x. For any subset F ⊂ E(M) denote by ddimxF the maximal number of independent functions from the set F at a point x ∈M . Put ddimF := max x∈M ddimxF . Definition 3.1. A bivector field (bivector for short) is a skew-symmetric morphism Π: T ∗M → TM . It is called Poisson if the operation {f, g}Π := Π(f)g is a Lie algebra on E(M) (here Π(f) := Π(df) stands for the Hamiltonian vector field corresponding to the function f). Define rank Π := max x∈M dim Π(T ∗xM) and RΠ := {x ∈M \| dim Π(T ∗xM) = rank Π}. A function f ∈ E(U) over an open set U ⊂M is called a Casimir function for a Poisson bivector Π if Π(f) ≡ 0. The set of all Casimir functions for Π over U will be denoted by ZΠ(U) (note that ZΠ(U) is the centre of the Lie algebra ( E(U), { , }Π ) ). Given a poisson bivector Π, the generalized distribution Π(T ∗M) ⊂ TM (called the charac- teristic distribution of Π) is integrable, the restrictions of Π to its leaves are correctly defined nondegenerate Poisson bivectors and the leaves are called the symplectic leaves of Π. In partic- ular the set RΠ is the union of all the symplectic leaves of maximal dimension. Definition 3.2. A set I ⊂ E(U) of functions over U ⊂ M is called involutive with respect to a Poisson bivector Π if {f, g}Π = 0 for any f, g ∈ I (we also say that such functions are in involution). An involutive set is complete with respect to Π if there exist f1, . . . , fs ∈ I, where s = dimM − 1 2 rank Π, independent at any point from some open dense set U0 ⊂ U . If I is a complete involutive set over U , then among fi there are dimM − rank Π Casimir functions of Π. Any such set I is a set of functions constant along a lagrangian foliation of dimension 1 2 rank Π defined on an open dense set in any symplectic leaf of maximal dimension. Definition 3.3. Two Poisson structures Π1 and Π2 on a manifold M is called compatible if Πt := t1Π1 + t2Π2 is a Poisson bivector for any t = (t1, t2) ∈ K2; the whole 2-dimensional family of Poisson bivectors (in case Π1,2 are linearly independent) {Πt}t∈R2 is called a bi-Poisson or a bi-Hamiltonian structure. 10 K. Lompert and A. Panasyuk Definition 3.4. A bi-Poisson structure {Πt} onM is Kronecker at a point x ∈M if rankC(t1Π1+ t2Π2)\|x is constant with respect to (t1, t2) ∈ C2\{0} (in the real analytic case we consider (Πj)x as a skew-symmetric bilinear form on the complexified cotangent space (T ∗xM)C). We say that {Πt} is Kronecker if it is Kronecker at any point of some open dense subset in M . Importance of this notion is explained by the following Theorem 3.5. Let {Πt} be a Kronecker bi-Poisson structure on M . Then for any open set U ⊂M such that ddimZΠt(U) = dimM − rank Πt for any t the set Z{Πt}(U) := Span (⋃ t6=0 ZΠt(U) ) is a complete involutive set of functions with respect to any Πt 6= 0 (see Definition 3.2). The reader is referred to [2] for the proof. The condition that ddimZΠt(U) = dimM−rank Πt for any t is always satisfied for any sufficiently small open set U and, in many cases also for an open and dense set in M . Remark 3.6. Recall that a real analytic submanifold M , dimRM = n, in a complex mani- fold M c, dimCM c = n, is called maximal totally real if in a neighbourhood of any point in M there exists a holomorphic coordinate system z = (z1, . . . , zn), zj = xj +iyj , such that M locally is given by the equations yj = 0. We say that M c is a complexification of M and M is real form of M c. The holomorphic coordinates as above will be called adapted to M . A complexification exists for any real analytic M [33]. Let M be a real analytic manifold and M c its complexification. Any real analytic tensor T defined on M can be uniquely extended to a holomorphic tensor T c defined in a vicinity of M in M c by extending its coefficients to holomorphic functions and substituting ∂ ∂xj and dxj by ∂ ∂zj and dzj respectively in the adapted systems of coordinates. Vice versa, if a holomorphic tensor T c is given on M c such that in the adapted coordinates its coefficients restricted to M are real, then it is the holomorphic extension of some real analytic tensor T on M . Obviously, if {Πt}t∈R2 , Πt = t1Π1 + t2Π2, is a real analytic bi-Poisson structure on M , then it is Kronecker at a point m ∈M if and only if so is its holomorphic extension {Πc}t∈C2 , Πc t = t1Πc 1 + t2Πc 2. Let G be a Lie group acting on a manifold M . Denote by EG(M) the space of all G-invariant functions from the set E(M). We say that a bi-Poisson structure {Πt} is G-invariant if so is each bi-vector Πt, t ∈ R2. Now we assume that the action of G on M is proper, as for instance is in the case of any smooth action of a compact Lie group. Fix some isotropy subgroup H ⊂ G determining the principal orbit type. In this case the subset MH = { x ∈M : Gx = gHg−1 for some g ∈ G } of M , consisting of all orbits G · x in M isomorphic to G/H, is an open and dense subset of M (see [8, Section 2.8 and Theorem 2.8.5]). It is well known that the orbit space M ′H := MH/G is a smooth manifold. There is a natural identification of spaces EG(MH) and p∗E(M ′H), where p : MH →M ′H = MH/G is the canonical projection, in particular ddimx EG(M) = ddim EG(M) for x ∈ MH . Moreover, if an G-invariant bi-Hamiltonian structure {Πt} is given on M , all the Poisson bivectors Πt\|MH are projectable with respect to p, i.e., there exist a correctly defined bi-Poisson structure {Π′t} on MH/G such that Π′t = p∗Πt, and the identification mentioned is a Poisson map: p∗{f, g}Π′t = {p∗f, p∗g}Πt , f, g ∈ E(M ′H). Invariant Nijenhuis Tensors and Integrable Geodesic Flows 11 Assuming that the reduced bi-Poisson structure {Π′t} is Kronecker, by Theorem 3.5 for a suffi- ciently small U ⊂M ′H we get an involutive family of functions Z{Π ′ t}(U) := Span (⋃ t6=0 ZΠ′t(U) ) , which is complete with respect to any Poisson structure Π′t. In some cases the corresponding set of functions p∗Z{Π ′ t}(U) on p−1(U) ⊂ M which by the considerations above is involutive with respect to any Poisson bivector Πt can be extended to a complete involutive set of functions. One such situation is touched in Theorem 3.7 below. This theorem also describes a method of proving the kroneckerity of the bi-Poisson structure {Π′t} reducing the problem to the calculation of rank of a finite number of the reduced Poisson structures, which was used in [17, 23]. Theorem 3.7. Retaining the assumptions above assume moreover that (a) the associated action ρ : g → Γ(TM) of the Lie algebra g of G on M can be extended to a holomorphic action ρc : gC → Γ(TM c) of the complexification gC of the Lie algebra g on some complexification M c of M on which a holomorphic extension {Πc t} of {Πt} is defined (see Remark 3.6) and {Πc t} is gC-invariant, i.e., the Lie derivative Lρc(ξ)Πc t is equal to zero for any t ∈ C2 and any ξ ∈ gC; here Γ(TM) stands for the space of real analytic vector fields on M and Γ(TM c) for the space of holomorphic vector fields on M c; (b) the action of G on M is generically locally free, i.e., the stabilizer H corresponding to the principal orbit type is finite; in particular, a generic stabilizer algebra of the actions ρ and ρc is trivial; (c) codim Sing g∗ ≥ 2, where Sing g∗ ⊂ g∗ is the union of the coadjoint orbits of nonmaximal dimension, i.e., Sing g∗ = g∗ \RΠg∗ for the Lie–Poisson structure Πg∗ on g∗; (d) for almost all t the bivector Πc t is nondegenerate and the action ρc is Hamiltonian with respect to Πc t , i.e., there exists a set E ⊂ C2 being the union of a finite number of 1- dimensional linear subspaces 〈t1〉, . . . , 〈ts〉, a map µct : M c → ( gC )∗ , t ∈ C2\E (the so-called momentum map), such that rank Πc t = dimM , t ∈ C2\E, and any fundamental vector field ρc(ξ), ξ ∈ gC, of this action is a Hamiltonian vector field Πc t ( Hξ t ) with the Hamiltonian function Hξ t (x) = 〈µct(x), ξ〉 and µct is a Poisson map from the Poisson manifold (M c,Πc t) to the Lie–Poisson manifold (( gC )∗ ,Π(gC)∗ ) ; (e) the restriction µt = µct \|M , t ∈ R2, takes values in g∗ ⊂ ( gC )∗ ; in particular the action ρ itself is Hamiltonian with respect to any Πt, t 6∈ R2 ∩ E: ρ(ξ) = Πt ( Hξ t \|M ) , ξ ∈ g. Then 1) the set U := MH \ (⋃ t∈R2 µ −1 t (Sing g∗) ) is an G-invariant open dense set in MH ; 2) the reduced bi-Poisson structure {Π′t} on M ′H = MH/G is Kronecker at a point x′ ∈ p(U) if and only if corank Π′ti \|x′ = ind g, i = 1, . . . , s; 3) if {Π′t} is Kronecker and F stands for any complete involutive set of polynomial functions on (g∗,Πg∗) (which exists by the Sadetov theorem [29]), the set of functions I := p∗ ( Z{Π ′ t}(M ′H) )⋃ µ∗t0F (3.1) is complete on MH with respect to any Πt0, t0 6∈ E ∩ R2; 12 K. Lompert and A. Panasyuk 4) moreover, p∗ ( Z{Π ′ t}(p(U)) ) = Span (⋃ t6=0 µ ∗ t ( ZΠg∗ (µt(U)) )) . Here ind g, the index of the Lie algebra g, is the codimension of a coadjoint orbit of maximal dimension, i.e., ind g = dim g− rank Πg∗. Proof. The G-invariance of the set U follows from the well-known fact that the Poisson property of the moment map µt is equivalent to its G-equivariance (with respect to the coadjoint action of G on g∗), which implies the G-invariance of µ∗t (Sing g∗). The so-called “bifurcation lemma” says that for any x ∈ MH the image (µt)∗(TxMH) ⊂ g∗ coincides with the annihilator in g∗ of the Lie algebra of the isotropy group Gx of x [22, Propo- sition 4.5.12]. Since this algebra vanishes by Assumption (b), rankµt(x) = dim g∗ and the image µt(MH) contains an open subset of g∗. The set Sing g∗ is algebraic and its complement in g∗ is open and dense, hence Assumption (c) guarantees that the set U is also open and dense. To prove item 2 observe that for any t 6= ti, i = 1, . . . , s and any x ∈ MH , by the holo- morphic version of the bifurcation lemma and by a simple algebraic fact (Lemma 3.8 below) corank((Πc t) ′)x = corank(Π(gC)∗)µct (x). Here ((Πc t) ′)x is the restriction of the bivector (Πc t)x treated as a bilinear skew-symmetric form on T ∗xM c to the annihilator (TxO)◦ ⊂ T ∗xM c of the tangent space TxO to the gC-orbit O passing through x and it is known that the space TxO is the skew-orthogonal complement to the tangent space through x of the fiber of the moment map µct . Hence, if moreover x ∈ U , then corankR(Π′t)p(x) = corankC((Πc t) ′)p(x) = ind gC = ind g. Therefore the reduced Poisson pencil {Πt} is Kronecker at p(x) if and only if the corank at p(x) of the reductions (Πti) ′ of the exceptional Poisson structures Πti , i = 1, . . . , s, is equal to ind g. Item 3 follows from the well known fact that once we have a pair of Poisson submersions p1 : (M,Π) → (M1,Π1) and p2 : (M,Π) → (M2,Π2) with skew-orthogonal fibers with respect to Π and complete families of functions F1, F2 on (M1,Π1), (M2,Π2) respectively, the family p∗1(F1) ∪ p∗2(F2) is complete on (M,Π) [23, Proposition 2.22]. The last item is a consequence of another well known fact that p∗1 ( ZΠ1 ) = p∗2 ( ZΠ2 ) [23, Corollary 2.19]. � Lemma 3.8. Let V be a vector space over K and ω : V ×V → K a nondegenerate skew-symmetric bilinear form. Denote by Π: V ∗ × V ∗ → K its inverse bivector. Let V1, V2 ⊂ V be two vector subspaces being orthogonal complements of each other with respect to ω. Then the restrictions of Π to the subspaces W1 := V ◦1 ⊂ V ∗ and W2 := V ◦2 ⊂ V ∗ have the same coranks. Proof. Indeed, since W1 and W2 are mutual orthogonal complements with respect to Π, we have ker(Π\|W1×W1) = W1 ∩W2 = ker(Π\|W2×W2). � 4 Bi-Poisson structures on cotangent bundles related to Nijenhuis (1, 1)-tensors Definition 4.1. Let Q be a manifold and X ∈ Γ(TQ) be a vector field on Q. Then the formula X̃ := Π ( X ) , where Π = ω−1 = ∂q ∧∂p is the canonical nondegenerate Poisson structure on T ∗Q inverse to the canonical symplectic form ω = dp∧dq and X stands for the linear function on T ∗Q corresponding to X, gives a vector field X̃ ∈ Γ(TT ∗Q) which will be called the cotangent lift ofX. The local characterization in the canonical (q, p)-coordinates is as follows: ifX = Xi(q) ∂ ∂qi , then X = Xi(q)pi and X̃ = Xi(q) ∂ ∂qi − pj ∂X j(q) ∂qi ∂ ∂pi . Remark 4.2. One can also describe the Hamiltonian function X as the evaluation θ ( X̃ ) of the canonical Liouville 1-form θ = pi dqi on X̃. Invariant Nijenhuis Tensors and Integrable Geodesic Flows 13 Remark 4.3. In particular, if a Lie group G with Lie(G) = g acts on a manifold Q and Xξ is the fundamental vector field of this action corresponding to an element ξ ∈ g, then X̃ξ is the corresponding fundamental vector field of the extended cotangent action of G on T ∗Q. Let (M,ω) be a symplectic manifold, ω1 another symplectic form on M . Then ω, ω1 are Poisson compatible (i.e., the Poisson structures Π := ω−1, Π1 := ω−1 1 are compatible) if and only if the (1, 1)-tensor Ñ := Π1◦ω : TM → TM is Nijenhuis (cf. [14, Proposition 7.1]). Assume this is the case. Let Πλ = Π1−λΠ. We have Πλ = ( Ñ−λI ) Π, therefore im Πλ = im ( Ñ−λI ) Π = im ( Ñ−λI ) since Π is nondegenerate. This gives us relation between characteristic distributions of Poisson structures and eigendistributions of a Nijenhuis tensor. In particular, we have proved the following lemma. Lemma 4.4. Retaining the assumptions above assume additionally that Ñ is semisimple and has constant eigenvalues λ1, . . . , λs, λi 6= λj, i 6= j (assumed to be real in the real category). Let D̃i, i = 1, . . . , s, be the eigendistribution corresponding to the eigenvalue λi. Then the foliation Fi tangent to the distribution ∑ j 6=i D̃j (which is integrable by Lemma 2.6) coincides with the symplectic foliation of the degenerate Poisson bivector Πλi. The following definition is due to F.-J. Turiel [32]. Definition 4.5. Let Q be a manifold and K : TQ→ TQ be a (1, 1)-tensor. Define its cotangent lift K̃ : TM → TM , M := T ∗Q, as follows. Let Kt : T ∗Q → T ∗Q be the map transposed to K understood as a smooth map M →M , let ω be the canonical symplectic form on T ∗Q, and let ω1 := ( Kt )∗ ω. Put K̃ := ω−1 ◦ ω1. If {qi} is a system of local coordinates onQ andK = Ki j(q) ∂ ∂qi ⊗dqj , then in the corresponding coordinates ( qi, pi ) on T ∗Q we have K̃ = Ki j(q) ( ∂ ∂qi ⊗ dqj + ∂ ∂pj ⊗ dpi ) + pk ( ∂Kk j ∂qi − ∂Kk i ∂qj ) ∂ ∂qj ⊗ dqi. Obviously, if K is a fiberwise invertible (1, 1)-tensor, then K̃−1 = K̃−1. Lemma 4.6 ([32]). TK = 0⇐⇒ T K̃ = 0. In particular, we have the following statement. Lemma 4.7. Let N : TQ→ TQ be a fiberwise invertible Nijenhuis (1, 1)-tensor. Then the pair of bivectors (Π,Π1), where Π := ω−1 is the canonical Poisson bivector on T ∗Q, Π1 := Ñ ◦ Π, and ω is the canonical symplectic form on T ∗Q, is a pair of compatible Poisson bivectors on T ∗Q. Proof. Obviously, Ñ ◦ Π = Ñ ◦ ω−1 = ( ω ◦ Ñ−1 )−1 = ( ω ◦ Ñ−1 )−1 = (( (N−1)t )∗ ω )−1 = N t ∗ω −1 = N t ∗Π, hence Π1 is a Poisson bivector. Since Ñ is a Nijenhuis tensor, Π and Π1 are compatible. � From now on we assume that N is an invertible semisimple Nijenhuis (1, 1)-tensor with con- stant eigenvalues λ1, . . . , λs, λi 6= λj , i 6= j (which are real in the real category) of multiplicities k1, . . . , ks respectively and let Di ⊂ TQ to be the eigendistribution corresponding to the eigen- value λi. We also denote by D̃i ⊂ TM the eigendistribution of the (1, 1)-tensor Ñ corresponding to the eigenvalue λi (of multiplicity 2ki). Let L(N) stand for the Lie algebra of vector fields on Q preserving N (i.e., V ∈ L(N) if and only if LVN = 0) and let θ = θ1 + · · ·+ θs be the decomposition of the canonical Liouville one-form θ on M := T ∗Q related to the decomposition TM = D̃1 ⊕ · · · ⊕ D̃s, i.e., θi\|D̃i = θ\| D̃i and θi ( ∑ j 6=i D̃j ) = 0. 14 K. Lompert and A. Panasyuk Lemma 4.8. Retain the assumptions above. Then the following statements hold. 1. For any i ∈ {1, . . . , s} the leaves of the symplectic foliation Fi of the Poisson bivector Πλi := Π1 − λiΠ are all of the same dimension and have codimension 2ki. For any such leaf F its image π(F ) under the canonical projection π : T ∗Q→ Q is a leaf of the foliation tangent to the distribution Ďi := ∑ j 6=i Dj (which is integrable by Lemma 2.6). The leaf π(F ) is of codimension ki in Q. For any leaf F0 of the foliation tangent to Ďi the set π−1(F0) is a Poisson submanifold of the Poisson manifold ( T ∗Q,Πλi ) . 2. The vector fields from L(N) tangent to the distribution Di form an ideal Li of the Lie algebra L(N) and there is a direct decomposition L(N) = L1 ⊕ · · · ⊕ Ls. 3. For any V ∈ L(N) and i ∈ {1, . . . , s} the function f iV := θi ( Ṽ ) ∈ E(T ∗Q), where Ṽ is the cotangent lift of V (see Definition 4.1), is a Casimir function of the bivector Πλi. In particular, since f iV linearly depends on V ∈ L(N), for any leaf F of the foliation Fi the formula V 7→ Φi F (V ) := f iV \|F defines a linear functional on L(N). 4. If V ∈ L(N) is tangent to the leaf π(F ), where F is a symplectic leaf of the Poisson bivector Πλi, then Φi F (V ) = 0. 5. Given any leaf F0 ⊂ Q of the foliation tangent to the distribution Ďi, a vector field V ∈ L(N) is tangent to F0 if and only if Ṽ is tangent to π−1(F0). Proof. Recall that ki is the multiplicity of the eigenvalue λi, i = 1, . . . , s. By Lemma 2.6 in a vicinity of every point on Q there exist a system of local coordinates ( qj 1 1 , . . . , q j1k1 , . . . , qj s 1 , . . . , qj s ks ) , where ( j1 1 , . . . , j 1 k1 , . . . , js1, . . . , j s ks ) = (1, . . . ,dimQ), such that the eigendistribution Di is spanned by the vector fields ∂ ∂qj i 1 , . . . , ∂ ∂q ji ki . Then Ñ = ∑ i λi ( ki∑ n=1 ( ∂ ∂qj i n ⊗ dqj i n + ∂ ∂p jin ⊗ dpjin )) and by Lemma 4.4 the tangent space to the symplectic foliation of Πλi is generated by the vector fields ∂ ∂ql , ∂ ∂pm , l,m 6∈ { ji1, . . . , j i ki } (the corresponding Casimir functions are qj i n , pjin , n = 1, . . . , ki). On the other hand, the tangent distribution to the leaves of ∑ j 6=i Dj is spanned by the vector fields ∂ ∂ql , l 6∈ { ji1, . . . , j i ki } . This proves the first assertion of the lemma. To prove item 2 notice that, given a vector field V = V l(q) ∂ ∂ql , the equality LVN = 0 holds if and only if [V,NX] = N [V,X] for any vector field X. Substituting X = ∂ ∂qj i n to the last equality we get λi [ V, ∂ ∂qj i n ] = N [ V, ∂ ∂qj i n ] , which means that [ V, ∂ ∂qj i n ] is an eigenvector of N corresponding to λi. Hence [ V, ∂ ∂qj i n ] is expressed as a linear combination of ∂ ∂qj i 1 , . . . , ∂ ∂q ji ki . In other words, the coefficients V ji1 , . . . , V jiki depend only on the coordinates qi := ( qj i 1 , . . . , q jiki ) for any i. For the proof of item 3 observe that θi = ki∑ n=1 pjin dqj i n in the coordinates mentioned and the evaluation f iV of this form on the cotangent lift Ṽ = V l(q) ∂ ∂ql − pl ∂V l(q) ∂qn ∂ ∂pn (4.1) of a vector field V ∈ L(N) is equal to f iV = ki∑ n=1 V jin ( qi ) pjin . Invariant Nijenhuis Tensors and Integrable Geodesic Flows 15 Any leaf F0 of the foliation tangent to Ďi is given in these coordinates by the equations qj i n = cj i n , n = 1, . . . , ki, and any symplectic leaf F ⊂ π−1(F0) of the foliation Fi by the equations qj i n = cj i n , pjin = Cjin , n = 1, . . . , ki, whose right hand sides are some constants. This proves item 3. If a vector field V ∈ L(N) is tangent to π(F ), then V jin ( ci ) = 0, n = 1, . . . , ki, where we put ci := ( cj i 1 , . . . , c jiki ) , in particular Φi F = f iV \|F = 0. The last item follows easily from formula (4.1). � Lemma 4.9. Retaining the assumptions of the preceding lemma assume that a transitive left action ρ : g → Γ(TQ), ξ 7→ Vξ, of a Lie algebra g on Q is given such that ρ preserves N , i.e., ρ(g) ⊂ L(N). Denote by ρ̃ the extended cotangent action, ρ̃(ξ) = ρ̃(ξ), ξ ∈ g. (Note that ρ is an antihomomorphism, the map V 7→ Ṽ is a homomorphism, hence ρ̃ is an antihomomorphism, i.e., a left action.) Given a leaf F of the symplectic foliation Fi, i ∈ {1, . . . , s}, let ϕiF := Φi F ◦ ρ ∈ g∗ be the linear functional induced on g by the functional Φi F ∈ (L(N))∗ from Lemma 4.8(3) and let gF stand for the stabilizer algebra of F , i.e., the set of elements ξ ∈ g such that ρ̃(ξ) is tangent to F . Let pi : Γ(TQ)→ Γ(Di) be the projection related to the decomposition TQ = D1⊕· · ·⊕Ds. Then 1) for any i ∈ {1, . . . , s} the map L(N)→ Lr induced by the projection pi is a homomorphism of Lie algebras; in particular ρi : g→ Γ(TQ), where ρi := pi ◦ ρ, is a left action of the Lie algebra g; 2) the action ρ̃ is Hamiltonian with respect to the Poisson structure Πλ for any λ 6= λi, i = 1, . . . , s with the momentum map µλ : T ∗Q→ g∗ given by 〈µλ(x), ξ〉 = (ψ(λ)∗θ)(ρ̃(ξ))(x), ξ ∈ g, (4.2) where θ is the canonical Liouville 1-form on T ∗Q and ψ(λ) is the diffeomorphism of T ∗Q given by ( (N − λI)t )−1 (we used the notation (·)t for the transposed map); equivalently, 〈µλ(x), ξ〉 = s∑ i=1 f i ρ(ξ) (x) λi−λ , see Lemma 4.8(3) for the definition of f iV ; moreover, µλ = µcan ◦ ψ(λ), (4.3) where µcan is the moment map corresponding to the canonical Poisson bivector Π; 3) given a leaf F of the symplectic foliation Fi, the restricted action of gF on F is Hamil- tonian with respect to the restriction of the Poisson structure Πλi to F with the mo- mentum map µFλi : F → g∗ given by 〈 µFλi(x), ξ 〉 = (ψ∗λiθ)(ρ̃(ξ))(x), ξ ∈ gF , where θ is the canonical Liouville 1-form on T ∗Q and ψλi is the smooth map of T ∗Q given by ψλj \|D∗i = ( (N − λiI)t )−1\|D∗i , j 6= i, ψλi \|D∗i = 0; here T ∗Q = D∗1 ⊕ · · · ⊕D∗s is the decom- position corresponding to TQ = D1 ⊕ · · · ⊕Ds; 3 4) the cotangent extension ρ̃i, ρ̃i(ξ) := ρ̃i(ξ), of the action ρi defined in item 1 is Hamiltonian with respect to the canonical Poisson bivector Π with the momentum map νi : T ∗Q → g∗, 〈νi(x), ξ〉 = f iρ(ξ)(x); 3Here an equivalent description of the momentum map similar to that from item 2 is also possible: 〈µFλi(x), ξ〉 = s∑ j=1 f j ρ(ξ) (x) λj−λi (note that the i-th term in the sum is correctly defined since f iρ̃(ξ) vanishes for ξ ∈ gF , cf. Lemma 4.8(4)). 16 K. Lompert and A. Panasyuk 5) for any leaf F0 ⊂ Q of the foliation tangent to the distribution Ďi its stabilizer algebra gF0 with respect to the action ρ, i.e., the set of ξ ∈ g such that ρ(ξ) is tangent to F0, coincides with the stabilizer algebra of the submanifold π−1(F0) with respect to the action ρ̃, i.e., the set of ξ ∈ g such that ρ̃(ξ) is tangent to π−1(F0); 6) the following inclusion holds: νi ( π−1(F0) ) ⊂ g⊥F0 ∼= (g/gF0)∗; 7) moreover, the relation F 7→ ϕiF = νi\|F is an gF0-equivariant one-to-one correspondence between the symplectic leaves F of Πλi such that π(F ) = F0 and linear functionals from a ki-dimensional linear subspace in (g/gF0)∗ (which in fact coincides with (g/gF0)∗, see Lemma 4.10(4)). 8) the stabilizer algebra gF ⊂ g of a leaf F ⊂ π−1(F0) with respect to ρ̃ is equal to the stabilizer algebra of the functional ϕiF ∈ (g/gF0)∗ with respect to the action of gF0. Proof. The claim of item 1 follows from the fact that each Li is an ideal in L(N) (see Lemma 4.8(2)). To prove items 2 and 3 use coordinates from the proof of the previous lemma. We have the following formulas: Π = ∑ i ki∑ n=1 ∂ ∂p jin ∧ ∂ ∂qj i n , Π1 = ∑ i λi ( ki∑ n=1 ∂ ∂p jin ∧ ∂ ∂qj i n ) , Πλ = ∑ i (λi − λ) ( ki∑ n=1 ∂ ∂p jin ∧ ∂ ∂qj i n ) , and Πλj = ∑ i 6=j (λi − λj) ( ki∑ n=1 ∂ ∂p jin ∧ ∂ ∂qj i n ) . For the vector field V = Vξ = ∑ i ki∑ n=1 V jin ξ ( qi ) ∂ ∂qj i n its cotangent lift Ṽ = Ṽξ takes the form Ṽ = ∑ i ki∑ n=1 V jin ξ ( qi ) ∂ ∂qjin − ∑ i ki∑ n,m=1 pjin ∂V jin ξ ( qi ) ∂qjim ∂ ∂pjim and is a Hamiltonian vector field with respect to Π: Ṽ = Π(Hξ), where Hξ = θ ( Ṽ ) =∑ i ki∑ n=1 V jin ξ ( qi ) pjin (cf. Remark 4.2). On the other hand, obviously, Ṽ = Πλ(Hλ ξ ), where we put Hλ ξ = ∑ i ki∑ n=1 V jin ξ ( qi ) pjin/(λi − λ). In fact, the functions Hλ ξ are global and correctly defined (i.e., they do not depend on the choices of local coordinates), which can be seen from the equality Hλ ξ = (ψ(λ)∗θ) ( Ṽξ ) . Yet another description of the function Hλ ξ is as follows: Hλ ξ = ∑ i f i Vξ /(λi − λ) (see Lemma 4.8(3) for the definition of f iVξ). Using the equality V[ξ,ζ] = [Vξ, Vζ ] we get Πλ ( Hλ ξ ) Hλ ζ −Hλ [ξ,ζ] = ∑ i ki∑ n,m=1 V jin ξ ( qi )∂V jim ζ ( qi ) ∂qjin pjim λi − λ − ∑ i ki∑ n,m=1 pjin ∂V jin ξ ( qi ) ∂qjim V jim ζ ( qi ) 1 λi − λ − ∑ i ki∑ n=1 V jin [ξ,ζ] ( qi ) pjin λi − λ = 0, which proves the hamiltonicity of ρ̃ with respect to Πλ, λ 6= λi. Formula (4.3) is a consequence of (4.2) as 〈µcan(x), ξ〉 = θ(ρ̃(ξ))(x). Invariant Nijenhuis Tensors and Integrable Geodesic Flows 17 Now assume that V = Vξ is tangent to the symplectic leaf F given in the local coordinates by the equations qj i n = const, pjin = const, n = 1, . . . , ki. Then by (4.1) we get Ṽ \|F = ∑ l 6=i kl∑ n=1 V jln ξ ( ql ) ∂ ∂qjln − ∑ l 6=i kl∑ n,m=1 pjln ∂V jln ξ ( ql ) ∂qjlm ∂ ∂pjlm = Πλi(Hξ) ∣∣ F , where Hξ = ∑ l 6=i kl∑ n=1 V jln ξ ( ql ) pjln/(λl − λi). The function Hξ is global and correctly defined for any ξ as Hξ = ∑ l 6=i f lVξ/(λl − λi) and Πλi(Hξ)Hζ −H[ξ,ζ] = ∑ l 6=i kl∑ n,m=1 V jln ξ ( ql )∂V jlm ζ ( ql ) ∂qjln pjlm λl − λi − ∑ l 6=i kl∑ n,m=1 pjln ∂V jln ξ ( ql ) ∂qjlm V jlm ζ ( ql ) 1 λl − λi − ∑ l 6=i kl∑ n=1 V jln [ξ,ζ] ( ql ) pjln λl − λi = 0. Since F is a Poisson submanifold with respect to Πλi , we have {Hξ\|F , Hζ \|F }Πλj \|F = {Hξ, Hζ}Πλj \|F = H[ξ,ζ]\|F . To prove item 4 notice that, if Vξ = ∑ i ki∑ n=1 V jin ξ ( qi ) ∂ ∂qj i n , ξ ∈ g, is the fundamental vector field of the action ρ, then ρi(ξ) = ki∑ n=1 V jin ξ ( qi ) ∂ ∂qj i n is the fundamental vector field of the action ρi. Its cotangent lift ρ̃i(ξ) is a Hamiltonian vector field with respect to Π with the Hamiltonian function H i ξ := f iρi(ξ) = f iVξ = ki∑ n=1 V jin ξ ( qi ) pjin . Now it remains to use the equality ρi([ξ, ζ]) = [ρi(ξ), ρi(ζ)], which implies Π ( H i ξ ) H i ζ −H i [ξ,ζ] = ki∑ n,m=1 V jin ξ ( qi )∂V jim ζ ( qi ) ∂qjin pjim − ki∑ n,m=1 pjin ∂V jin ξ ( qi ) ∂qjim V jim ζ ( qi ) − ki∑ n=1 V jin [ξ,ζ] ( qi ) pjin = 0. Item 5 follows from Lemma 4.8(5) and item 6 follows from Lemma 4.8(4) in view of the fact that π−1(F0) is foliated by the symplectic leaves of the Poisson bivector Πλi (see Lemma 4.8(1)) and from the equality Φi F (ρi(ξ)) := f iρi(ξ)\|F , where F is any such leaf. To prove item 7 first notice that the gF0-equivariance follows from g-equivariance of the moment map νi. Now recall (see the proof of Lemma 4.8) that ϕiF (ξ) = ki∑ n=1 V jin ξ ( ci ) Cjin , ξ ∈ g, where the constants ci, Cjin specify the particular leaf F and V jin ξ are the coefficients of the fundamental vector field ρ(ξ) = V l ξ (q) ∂ ∂ql . 18 K. Lompert and A. Panasyuk Now fix a leaf F0 of the foliation tangent to a distribution Ďi, i.e., fix constants ( ci ) . For any ξ ∈ g we have a linear map4 ( Cji1 , . . . , Cjiki ) 7→ ki∑ n=1 V jin ξ ( ci ) Cjin , expressing the correspondence F 7→ ϕiF (ξ), where ki = corank Ďi. Thus the claim of item 7 is equivalent to the nondegeneracy of the following matrix V ji1 ξ1 ( ci ) · · · V jiki ξ1 ( ci ) ... ... V ji1 ξki ( ci ) · · · V jiki ξki ( ci )  , where ξ1, . . . , ξki ∈ g are linearly independent elements not belonging to gF0 . In turn, the nondegeneracy of this matrix follows from the fact that g acts transitively on G/K and, as a consequence, on the space of leaves of the foliation tangent to the distribution Ďi. Finally the last item follows from item 7. � Now we apply the preceding results to homogeneous spaces. Let G/K be a homogeneous space and let N be an G-invariant semisimple Nijenhuis (1, 1)-tensor on G/K with the real spectrum {λ1, . . . , λs}. Then by Theorem 2.7 there exists a decomposition g = g1 + · · ·+ gs to the sum of subspaces such that 1) ∀i,j∈{1,...,s},i 6=j gi ∩ gj = k; 2) ∀i,j∈{1,...,s} gi + gj are Lie subalgebras in g; 3) the decomposition above induces the decomposition T (G/K) = D1⊕· · ·⊕Ds to integrable subbundles and N \|Di = λi IdDi . Write P : G→ G/K and π : T ∗(G/K)→ G/K for the canonical projections. By the construction from the proof of Lemma 2.5 the eigendistribution Di of N corresponding to the eigenvalue λi is equal P∗D̂i, where D̂i is the left invariant distribution on G obtained from the subspace gi ⊂ g ∼= TeG. In particular, since kerP∗ is the left invariant distribution obtained from the subspace k ⊂ gi ⊂ g ∼= TeG, the rank of Di, i.e., the multiplicity ki of the eigenvalue λi, is equal to dim(gi/k). Denote ǧi := ∑ j 6=i gj (this is a Lie subalgebra in g by condition 2) and let Ǧi be the corre- sponding subgroup in G. By Lemma 2.5 the leaves of the foliation integrating the distribution Ďi := ∑ j 6=i Dj are the projections with respect to P of the left cosets gǦi, g ∈ G. Let pi : TQ→ Di be the projection related to the decomposition TQ = D1 ⊕ · · · ⊕Ds. Lemma 4.10. Let N be an invertible Nijenhuis (1, 1)-tensor on a homogeneous space Q = G/K satisfying the assumptions above. Let Π be the canonical poisson bivector on T ∗(G/K) and Π1 = Ñ ◦Π (see Lemma 4.7). Then 1) for any symplectic leaf F of the Poisson bivector Πλi := Π1 − λiΠ there exists an element g ∈ G such that π(F ) = P ( gǦi ) ; such element g is unique modulo right multiplication by h ∈ Ǧi; 4Note that although the range of constants ci is bounded by that of the local coordinates qi, the constants Cjin can take any value. Invariant Nijenhuis Tensors and Integrable Geodesic Flows 19 2) the stabilizer algebra gπ(F ) ⊂ g of the leaf π(F ) = P ( gǦi ) of the foliation tangent to the distribution Ďi with respect to the G-action on G/K is equal to Adg ǧi; 3) the stabilizer algebra gF ⊂ g of the leaf F with respect to the extended G-action on T ∗(G/K) is equal to the stabilizer algebra gϕ i F ⊂ Adg ǧi of the functional ϕiF ∈ (g/Adg ǧi) ∗ con- structed in Lemma 4.9 by means of an action ρ, where we specify ρ : g → Γ(T (G/K)) to be the natural action of the Lie algebra g on G/K; 4) if F0 ⊂ G/K is a fixed leaf of the foliation tangent to the distribution Ďi, F0 = P ( gǦi ) (g fixed), the relation F 7→ ϕiF is an Adg ǧi-equivariant one-to-one correspondence be- tween the symplectic leaves F of Πλi such that π(F ) = F0 and linear functionals from (g/Adg ǧi) ∗. Proof. First and second items are consequences of Lemma 2.5 applied to the subalgebra h = ǧi. Item 3 follows from item 2 and Lemma 4.9(8). Item 4 follows from Lemma 4.9(7) since ki = dim gi − dim k = dim(g/Adg ǧi) ∗. � 5 Algebraic criterion of kroneckerity in the case of a locally free action The theorem below is the main result of this paper. Let G be a compact Lie group, K its closed subgroup. Assume that the natural action of G on M = T ∗(G/K) is generically locally free, i.e., the stabilizer corresponding to the principal orbit type is finite. Fix such a stabilizer H. In this case the subset MH = { x ∈M : Gx = gHg−1 for some g ∈ G } of M , consisting of all orbits G · x in M isomorphic to G/H, is an open and dense subset of M and the orbit space M ′H := MH/G is a smooth manifold (cf. Section 3). Write p : MH →MH/G for the canonical projection. Theorem 5.1. Let N be an G-invariant invertible5 semisimple Nijenhuis (1, 1)-tensor on G/K with the real spectrum {λ1, . . . , λs}, i.e., (cf. Theorem 2.7) there exists a decomposition g = g1 + · · ·+ gs (5.1) to the sum of subspaces such that • ∀i,j∈{1,...,s},i 6=j gi ∩ gj = k; • ∀i,j∈{1,...,s} gi + gj are Lie subalgebras in g; • the decomposition above induces the decomposition T (G/K) = D1 ⊕ · · · ⊕Ds to integrable subbundles and N \|Di = λi IdDi. Let (Π,Π1) be the Poisson pair consisting of the canonical Poisson bivector Π on T ∗(G/K) and of the Poisson bivector Π1 = Ñ ◦ Π, where Ñ is the cotangent lift of the (1, 1)-tensor N (see Definition 4.5 and Lemma 4.7). Then the bi-Poisson structure generated by the reduced Poisson pair (p∗Π, p∗Π1) is Kronecker at any point of the set p(W ), where W ⊂MH is the open dense set which will be specified in the proof, if and only if for any i = 1, . . . , s ∃ ai ∈ (g/ǧi) ∗ : ind gai + codim(g/ǧi)∗ Oai = ind g, (5.2) 5Invertibility can be always achieved by adding the identity operator, which does not change the corresponding pencil of operators and the related Poisson pencil. 20 K. Lompert and A. Panasyuk where ǧi = ∑ j 6=i gi and gai and Oai are respectively the stabilizer algebra and the orbit of the element ai with respect to the coadjoint action ad∗ : ǧi → gl((g/ǧi) ∗). Equivalently, condition (5.2) can be written as ind(ǧi n (g/ǧi)) = ind g, (5.3) where the term in the l.h.s. is the semidirect product of the Lie algebra ǧi and the vector space (g/ǧi) with respect to the ad-action. Proof. We first note that conditions (5.2) and (5.3) are equivalent by Lemma 5.2 below. Let U = MH \ (⋃ λ µ −1 λ (Sing g∗) ) , where the moment map µλ is specified in Lemma 4.9(2). Observe that all the objects involved admit a natural complexification (cf. Remark 3.6): the compact Lie groups G and K are imbedded in their Chevalley complexifications Gc and Kc and the homogeneous space Q = G/K is imbedded into the complex homogeneous space Qc = Gc/Kc. Moreover, the decomposition (5.1) implies the decomposition gC = gC1 + · · ·+ gCs , which in turn induces the decomposition TQc = Dc 1⊕ · · · ⊕Dc s of the holomorphic tangent bun- dle to Qc to complex analytic involutive distributions and a complex analytic (1, 1)-tensor N c given by N \|Dci = λi IdDci . By Lemma 4.9(2) the assumptions of Theorem 3.7 are satisfied (it is well-known that for reductive Lie algebras codim Sing g∗ ≥ 3) and we conclude that the reduced bi-Poisson structure {( Πλ )′} , ( Πλ )′ = p∗Π1 − λp∗Π, is Kronecker at a point x′ ∈ p(U) if and only if corank p∗Π λi \|x′ = ind g, i = 1, . . . , s, where Πλi = Π1− λiΠ (see Theorem 3.7(2)). Below we express the number corank p∗Π λi \|x′ in equivalent terms, see formula (5.4). From Lemma 4.9(3) it follows that the restriction of the action ρ̃ : g→ Γ(T (T ∗G/K)) to the stabiliser subalgebra gF of any symplectic leaf F of the Poisson bivector Πλi is Hamiltonian with respect to this bivector with the momentum map µFλi : F → g∗. Obviously the action of gF is also locally free. Therefore by the bifurcation lemma (cf. the proof of Theorem 3.7) the corank of the reduction ( Πλi \|F )′ of the Poisson structure restricted to the symplectic leaf, Πλi \|F , at the point x′ = p(x), where x ∈ F , is equal to the index of the Lie algebra of gF , provided µFλi(x) 6∈ Sing gF . The algebraic set Sing gF is nowhere dense in g∗F and the set UF = (F ∩ U) \ (⋃s i=1 ( µFλi )−1) (Sing gF ) is an open dense set in F and, moreover, V = ⋃ F UF is open and dense in MH . From now on we will consider only points x′ ∈ p(V ). Obviously, corank(MH/G) ( Πλi )′ x′ = corankF/GF ( Πλi \|F )′ x′ + codimSi G · F . Here GF is the subgroup in G corresponding to the subalgebra gF , Si stands for the space of symplectic leaves of the Poisson bivector Πλi , on which a natural action of the group G is induced from the action of G on M due to the G-invariance of Πλi , and G·F denotes the orbit of the point F ∈ Si with respect to this action. Recall (see Lemma 4.8(1)) that the space Si is foliated by the submanifolds of the form π−1(F0), where F0 ⊂ G/K is a leaf of the foliation tangent to the distribution Ďi = ∑ j 6=i Dj . Since the group G acts transitively on G/K and as a consequence on the space of leaves of the foliation tangent to the distribution Ďi, we have codimSi G · F = codimSi\|π−1(π(F ))Gπ(F ) · F , where Gπ(F ) is the subgroup corresponding to the subalgebra gπ(F ), i.e., the stabilizer of the sub- manifold π−1(π(F )) with respect to the cotangent action (see Lemma 4.9(5)) and Si\|π−1(π(F )) stands for the submanifold in Si of leaves contained in the Poisson submanifold π−1(π(F )). In view of Lemma 4.9(7), Lemma 4.10(4) and Lemma 4.9(8) Si\|π−1(π(F )) can be identified with (g/gπ(F )) ∗, Gπ(F ) · F with OϕiF and gF with gϕ i F , where ϕiF ∈ (g/gπ(F )) ∗ is the functional cor- responding to F and gϕ i F and OϕiF are respectively its stabilizer and orbit with respect to the action of gπ(F ) on (g/gπ(F )) ∗. Thus we have proven that corank(MH/G) ( Πλi )′ x′ = corankF/GF ( Πλi \|F )′ x′ + codimSi G · F = ind gF + codimSi\|π−1(π(F ))Gπ(F ) · F = ind gϕ i F + codim(g/gπ(F )) ∗ OϕiF . Finally, in view of Lem- Invariant Nijenhuis Tensors and Integrable Geodesic Flows 21 ma 4.10(2), we have gπ(F ) = Adg ǧi for some g ∈ G and corank ( Πλi )′ x′ = ind gϕ i F + codim(g/Adg ǧi)∗ OϕiF . (5.4) We are ready to finish the proof. Assume that {( Πλ )′} is Kronecker at x′. Then by Theo- rem 3.7 ind gϕiF + codim(g/Adg ǧi)∗ OϕiF = ind g, i ∈ {1, . . . , s}. Acting by Adg−1 we will get condition (5.2). Vice versa, assume that (5.3) is satisfied. Then by the Räıs formula (see Lemma 5.2) ind g = ind gai + codim(g/ǧi)∗ Oai for ai ∈ R((g/ǧi) ∗). Obviously also ind g = ind gg·ai + codim(g/Adg ǧi)∗ Og·ai and g · ai ∈ R((g/Adg ǧi) ∗) for any g ∈ G, where g · ai := Ad∗g−1 ai. Fix g ∈ G and let Fi be the symplectic leaf of the Poisson bivector Πλi corresponding to the ele- ment g · ai by Lemma 4.10(4) (with π(Fi) = P ( Adg Ǧi ) ). Note that the leaves Fi are mutually transversal and ∑ i codimFi = dimM , thus ⋂ i Fi is a point, say x. Recall (see Lemma 4.9(6), (7) and Lemma 4.10(4)) that the map νgi := νi\|π−1(π(Fi)) : π−1(π(Fi))→ g⊥π(Fi) ∼= (g/Adg ǧi) ∗ is an epimorphism. The set ( νgi )−1 (R((g/Adg ǧi) ∗) is an open dense set in π−1(π(Fi)) and the set W = V ∩ (⋃ g∈G ⋂s i=1 ( νgi )−1) (R((g/Adg ǧi) ∗)) is an open dense set in MH . Taking ai such that g · ai ∈ νgi ( W ∩ π−1(π(Fi)) ) for any i, we achieve that x ∈ W ⊂ V . Formula (5.4) shows that ind gg·ai + codim(g/Adg ǧi)∗ Og·ai = corank p∗Π λi \|x′ , where x′ = p(x). By Theorem 3.7 we conclude that {(Πλ)′} is Kronecker at x′. � Lemma 5.2. Let g be a Lie algebra and h ⊂ g its Lie subalgebra. Then the condition of existing a ∈ (g/h)∗ such that ind ha + codim(g/h)∗ Oa = ind g, (5.5) where ha and Oa are respectively the stabilizer algebra and the orbit of the element a with respect to the coadjoint action ad∗ : h→ gl((g/h)∗), is equivalent to the following one: ind(hn (g/h)) = ind g, (5.6) where the Lie algebra in the l.h.s. is the semidirect product of the Lie algebra h and the vec- tor space (g/h) with respect to the ad-action. Moreover, if one of this condition holds, the equality (5.5) holds for any a from the open dense set R((g/h)∗) ⊂ (g/h)∗, R((g/h)∗) := {α ∈ (g/h)∗ \| ∃β ∈ (hn (g/h))∗ \ Sing((hn (g/h))∗) : α = β\|(g/h)}. Proof. Recall [28, Proposition 1.3(i)] that ind(hn (g/h)) = ind ha + codim(g/h)∗ Oa for a ∈ R((g/h)∗). Hence (5.6) implies (5.5). On the other hand, for arbitrary a we have ind ha + codim(g/h)∗ Oa = corank Π(hn(g/h))∗ \|β ≥ ind(h n (g/h)), where β ∈ Sing(h n (g/h))∗ is any element with a = β\|(g/h) (cf. [25, Theorem 1.1]) and, moreover, the number ind(hn (g/h)) is bounded below by ind g (since hn (g/h) is a contraction of g). Thus, if ind ha+codim(g/h)∗ Oa = ind g for some a, then ind(hn (g/h)) = ind g. � Remark 5.3. In the case when K = {e} is the trivial subgroup of the Lie group G, condi- tion (5.2) coincides with the necessary and sufficient condition of kroneckerity of the Lie–Poisson pencil related to an algebraic Nijenhuis operator obtained in [24, Theorem 2.5]. 22 K. Lompert and A. Panasyuk Theorem 5.4. Retain the assumptions of Theorem 5.1 and assume that one of the equivalent conditions (5.2), (5.3) hold. Let b be the G-invariant metric on G/K, called normal, induced by some biinvariant metric on G, i.e., by an AdG-invariant bilinear form B on g. Then the G-invariant metric bN , bN (x, y) := b((N +N∗)x, y), x, y ∈ Γ(T (G/K)), on G/K corresponding to the symmetric (1, 1)-tensor N +N∗, where N∗ is the adjoint to N (1, 1)-tensor, b(N∗x, y) = b(x,Ny), as well as the normal metric itself have completely integrable geodesic flows in the class of analytic integrals polynomial in momenta. Proof. It is well-known that a function of the form µ∗canf , where µcan is the moment map of the G-action on T ∗(G/K) corresponding to the canonical Poisson bivector Π and f is any polynomial on g∗, is analytic and polynomial in momenta. Indeed, the analyticity is obvious and the polynomiality can be argued as follows. If ξ ∈ g is treated as a linear function on g∗, the function Hξ = µ∗canξ is the Hamiltonian function of the corresponding fundamental vector field Vξ, which in turn can be treated as a fiberwise linear function on T ∗(G/K) (cf. Definition 4.1 and Remark 4.3). Thus, if f is a polynomial in ξ, then µ∗canf is fiberwise polynomial. By Theorem 3.7(3) the involutive set of functions I (3.1), where Πt0 = Π, is complete on T ∗(G/K). We have to prove that the quadratic forms q(x) := b(x, x) and qN (x) := bN (x, x), x ∈ Γ(T ∗(G/K)), where we identified T (G/K) with T ∗(G/K) by means of b, is contained in this set. Let Q(x) = B(x, x) be the quadratic form of B understood as a Casimir function on g∗ after the identification of g and g∗ by means of B. Then by Theorem 3.7(4) the function µ∗canQ belongs to I. One can show that in fact µ∗canQ coincides with q. Indeed, b belongs to the class of the so-called submersion metrics obtained from the right-invariant metrics on G by the canonical submersion G→ G/K. The quadratic forms of all the submersion metrics are of the form µ∗canf , where f is the corresponding quadratic polynomial on g ∼= g∗ [6, Section 7]. To prove that qN ∈ I recall that by Theorem 3.7(4) the set I besides the functions µ∗canF consists of the functions of the form µ∗λf , λ 6= λi, f ∈ ZΠ. On the other hand, µλ = µcan ◦ ((N − λI)−1)t by formula (4.3), hence the functions of the form b ( ((N − λI)−1)∗x, ( (N − λI)−1 )∗ x ) = b((N − λI)−1 ( (N − λI)−1 )∗ x, x), x ∈ Γ(T ∗(G/K)), belong to I. Moreover, I will contain also the coefficients of the Laurent expansion b ( (N−λI)−1 ( (N−λI)−1 )∗ x, x ) = 1 λ2 b(x, x)+ 1 λ3 b((N+ N∗)x, x) + · · · corresponding to the expansion (N − λI)−1 = − ( 1 λI + 1 λ2 N + · · · ) . Finally, the functions µ∗λf , where f are polynomial Casimir functions of Π, are polynomial in momenta (since ( (N − λI)−1 )t is a fiberwise linear map). � 6 Applications: two homogeneous spaces with integrable geodesic flows In the table from Example 2.12 among the triples (g, g1, g2) of compact Lie algebras such that g = g1+g2 of one can find two distinguished from our point of view series: (A2n−1, Cn, A2n−2⊕T ) and (Dn+1, Bn, An ⊕ T ). For both of them the pairs (g, gi), i = 1, 2, are symmetric, i.e., the Lie algebra gi is the fixed point set of an automorphism of g of second order (cf. [11, Tables II, III, Section 6, Chapter X]). In this context we have to mentioned the following reformulation of the result of Brailov [31, Theorem 5, Section 37]. Theorem 6.1. Let g be a semisimple Lie algebra and g0 ⊂ g its symmetric subalgebra. Then ind g = ind(g0 n (g/g0)), where g0 n (g/g0) is the so-called Z2-contraction of g, i.e., the semidirect product of the Lie algebra g0 and the vector space g/g0 with respect of the natural ad-representation of g0 in g/g0. Invariant Nijenhuis Tensors and Integrable Geodesic Flows 23 In particular, it follows from this result that both the series of decompositions g = g1 + g2 mentioned satisfy condition (5.3) of Theorem 5.1. This allows us to formulate the following theorem. Theorem 6.2. Let G/K be one of the following homogeneous spaces: (a) SU(2n)/(S(U(2n− 1)×U(1)) ∩ Sp(n)); (b) SO(2n+ 2)/(SO(2n+ 1) ∩U(n+ 1)). Then the geodesic flow of 1) the normal metric b on G/K and 2) the G-invariant metric bN (see Theorem 5.4) corresponding to the G-invariant Nijenhuis (1, 1)-tensor N on G/K with the real spectrum {λ1, λ2}, λ1 6= λ2, λi 6= 0, related to the decomposition g = g1 + g2 with k = g1 ∩ g2 by Theorem 2.7 is completely integrable in the class of analytic integrals polynomial in momenta. Here g and k are the Lie algebras of G and K respectively and the triples of subalgebras (g, g1, g2) are equal to (A2n−1, Cn, A2n−2⊕ T ) and (Dn+1, Bn, An⊕ T ) respectively. The explicit formulae for the embeddings gi ⊂ g as well as the decomposition k⊥ = k1⊕k2 of the complementary to k space corresponding to the decomposition g = g1 +g2 and the “inertia operator” nk⊥+n∗ k⊥ = (N +N∗)\|To(G/K)∼=k⊥ (here o = P (e)) are listed in Appendix A. Proof. In view of Theorem 6.1 the result will follow form Theorem 5.4 if we ensure that the action of G on T ∗(G/K) is locally free (which is an essential assumption of Theorem 5.4). Below we prove this fact, which is equivalent to the fact that the stabilizer kE := stabρ(E) of a generic element E in k⊥ under the isotropy action ρ : k → gl ( k⊥ ) vanishes; here k⊥ is the orthogonal complement to k with respect to the (nondegenerate) Killing form on g and we identify isotropy and coisotropy action by means of this form restricted to k⊥. In other words, kE coincides with Zk(E), the centralizer of the element E ∈ k⊥ in k. In fact, since the function dim kE is lower semicontinuous, it is enough to show the existence of an element E with kE = {0}. Note that it is sufficient to show the existence of such an element for the complexified action which we do below. We list explicit realizations of the complexifications ( gC, gC1 , g C 2 ) for the above mentioned triples (g, g1, g2) as well as the subspace ( kC )⊥ complementary to the subspace kC = gC1 ∩gC2 with respect to the Killing form. Besides, we indicate the element E ∈ ( kC )⊥ with stabρ(E) = {0} and outline the proof of the last equality. We consider separately cases (a) and (b).6 Case (a): (a2n−1, cn, a2n−2 ⊕ t). gC = sl(2n,C), gC1 = {[ Z1 Z2 Z3 −Z1 T ] \|Z2 = ZT2 , Z3 = ZT3 } ∼= sp(n,C), gC2 = {[ Z 0 0T −t ] \|Z ∈ gl(2n− 1,C), t = TrZ } ∼= gl(2n− 1,C), kC =   Ã 0 B̃ 0 0T t 0T 0 C̃ 0 −ÃT 0 0T 0 0T −t  \| Ã, B̃, C̃ ∈ gl(n− 1,C), B̃ = B̃T , C̃ = C̃T , t ∈ C  ∼= sp(n− 1,C)⊕ t, 6We switch to modern notations and denote the classical Lie algebras by small Gothic letters. 24 K. Lompert and A. Panasyuk ( kC )⊥ =   A v1 B v2 uT1 a uT2 b C v3 AT v4 uT3 c uT4 a  \|vi,ui ∈ Cn−1, B = −BT , C = −CT , a = −TrA, b, c ∈ C  . The isotropy action ρ : k → gl ( k⊥ ) can be decomposed into direct sum of two invariant subspaces V1 :=   A 0 B 0 0T 0 0T b C 0 AT 0 0T c 0T 0   , V2 :=   0 v1 0 v2 u1 0 u2 0 0 v3 0 v4 u3 0 u4 0   , and a trivial 1-dimensional representation which will be neglected. Let ρ : k→ gl(V1 ⊕ V2) be the coisotropy representation with the invariant subspaces Vi and let Ei ∈ Vi. Then obviously stabρ(E1 + E2) = stabρ̃(E2), where ρ̃ := ρ\|stabρ(E1). Take the element k⊥ 3 E1 =  A 0 0 0 0 0 0 1 0 0 AT 0 0 0 0 0  , with A = [ 0 1 . . . . . . 1 0 ] , the standard nilpotent matrix. Then stabρ(E1) consists of the matrices of the form Y =  A 0 B 0 0 0 0 0 C 0 −AT 0 0 0 0 0  , where A =  a1 a2 . . . an−1 a1 . . . ... . . . a2 a1  , B =  b1 . . . bn−2 bn−1 ... ... bn−1 bn−2 ... bn−1  , C =  c1 c1 c2 ... ... ... c1 c2 . . . cn−1  . Invariant Nijenhuis Tensors and Integrable Geodesic Flows 25 Choose E2 ∈ V2 with u1 = vT1 = (1, . . . , 1) and trivial vi, ui, i > 1. Then for Y ∈ stabρ(E1) we have [E2, Y ] =  0 u 0 0 v 0 w2 0 0 w1 0 0 0 0 0 0  , where u =  a1 + · · ·+ an−2 + an−1 a1 + · · ·+ an−2 ... a1  , v = −  a1 a1 + a2 ... a1 + a2 + · · ·+ an−1  T , w1 = −  c1 + c2 + · · ·+ cn−1 c2 + · · ·+ cn−1 ... cn−1  , w2 =  b1 + b2 + · · ·+ bn−1 b2 + · · ·+ bn−1 ... bn−1  T . If Y ∈ stabρ̃(E2), then ai = 0, bi = 0, and ci = 0, which implies stabρ(E) = {0}, where E := E1 + E2. Case (b): (dn+1, bn, an ⊕ t). gC = {[ Z1 Z2 Z3 −Z1 T ] \|Zi ∈ gl(n+ 1,C), Z2 = −ZT2 , Z3 = −ZT3 } ∼= so(2n+ 2,C), gC1 =   0 uT 0 vT −v W1 −v W2 0 uT 0 vT −u W3 −u −W T 1 \|W2 = −W T 2 , W3 = −W T 3 , v,u ∈ Cn ∼= so(2n+ 1,C), gC2 = {[ Ã 0 0 −ÃT ] ∈ g \| Ã ∈ gl(n+ 1,C) } ∼= gl(n+ 1,C), kC = Y :=  0 0T 0 0T 0 A 0 0 0 0T 0 0T 0 0 0 −AT  \|A ∈ gl(n,C)  ∼= gl(n,C), ( kC )⊥ = Z :=  a uT −v 0 Z2 Z3 −a vT −u 0  \|Z2 = −ZT2 , Z3 = −ZT3 , u,v ∈ Cn, a ∈ C  . For Y ∈ kC and Z ∈ ( kC )⊥ as above with Z2 = [ 0 wT 2 −w2 B ] , Z3 = [ 0 wT 3 −w3 C ] , where B and C are skew-symmetric n× n matrices, one has [Y,Z] =  0 −uTA 0 (Aw2)T Av 0 −Aw2 AB +BAT 0 −wT 3 A 0 −(Av)T ATw3 −(ATC + CA) ATu 0  . 26 K. Lompert and A. Panasyuk We will prove the triviality of the stabilizer of the element E = [ 0 J J 0 ] ∈ k⊥, where J = 0 1 −1 . . . 1 −1 0 . Observe that conditions Aw2 = 0, ATw3 = 0 for w2 = w3 = (1, 0, . . . , 0)T imply that a1,i = 0, ai,1 = 0, where we put A = \|\|aij \|\|ni,j=1. Thus for any Y ∈ stab(E) the matrix A is of the form [ 0 0 0 An−1 ] where An−1 ∈ gl(n − 1,C). Next, such Y ∈ stab(E) if and only if simultaneously AB +BAT =  0 0 0 0 0 0 0 0 a22 a32 . . . an,2 0 −a22 0 ∗ ∗ ∗ 0 −a32 ∗ 0 ∗ ∗ 0 ... ∗ ∗ ∗ 0 −an,2 ∗ ∗ ∗ 0  = 0 and ATC + CA =  0 0 0 0 0 0 0 0 a22 a23 . . . a2,n 0 −a22 0 ∗ ∗ ∗ 0 −a23 ∗ 0 ∗ ∗ 0 ... ∗ ∗ ∗ 0 −a2,n ∗ ∗ ∗ 0  = 0. Therefore A has to be of the form [ 0 0 0 An−2 ] with An−2 ∈ gl(n − 2,C). By induction we conclude that A = 0, i.e., stab(E) = {0}. � 7 Concluding remarks We would like to note that in the proof of Theorem 5.1 we tried to maximally accurately indicate the open dense set W such that the reduced bi-Poisson structure is Kronecker at any point of p(W ) (and is not Kronecker in the complement). This is important from the point of view of study qualitative analysis of the geodesic flow since outside this set the singularities of the corresponding lagrangian fibration appear (cf. [4]). The assumption of compactness of the Lie groups G and K which appeared in Theorem 5.1 (see also Theorem 3.7) was used in order to guarantee (1) the existence of complexification of the homogeneous space G/K and as a consequence of other related objects; (2) the existence of an G-invariant open dense set M0 in M = T ∗(G/K) (the set MH) such that the orbit space M0/G is a smooth manifold. In fact, the assumption of compactness can be essentially weakened (since conditions (1) and (2) can be achieved for a wider class of Lie groups) preserving the conclusion of the theorem. We did not discuss these weaker assumptions as the main application (Theorem 6.2) is aimed in the class of compact homogeneous spaces. The assumption that the action of G on T ∗(G/K) is free, which is essential in Theorems 3.7 and 5.1, can be bypassed by a special reduction to smaller groups instead of G and K, see [18] and [17]. We would like to mention that a matter of further research is the study of the cases when the necessary and sufficient conditions (5.3) are not satisfied. In such cases the canonical commuting set of functions B related to the reduced bi-Hamiltonian structure is not complete. However, Invariant Nijenhuis Tensors and Integrable Geodesic Flows 27 based on the experience from the study [26] of bi-Hamiltonian structures related with Lie pen- cils (hence in fact reductions of (T ∗(G/K),Π,Π1) with trivial K) one could expect additional symmetries in this case and, as a consequence, additional Noether integrals. One can ask for algebraic conditions sufficient for the completeness of the family B enlarged by these integrals. Finally, it is worth mentioning that our theory related to the triples (g, g1, g2) (see Section 6) is very close to that appearing within the generalized chain method [5, 6]. Note however, that the assumption of maximality of rank of the symmetric space, which is essential in [6, Theorem 8.6], is not satisfied for our symmetric pairs (g, gi), i.e., the overlap between the theories mentioned is minimal (and requires further study). A Compact real forms of the triples (g, g1, g2) and inertia operators Below we list explicit realizations of the compact real forms for the triples (g, g1, g2) used in Theorem 6.2 as well as the decompositions of the subspace k⊥ = k1 ⊕ k2 complementary to the subspace k with respect to the Killing form induced by the decompositions g = g1 + g2, where ki = gi∩k⊥, and formulae for the “inertia operators” nk⊥+n∗ k⊥ : k⊥ → k⊥ induced by the operator nk⊥ : k⊥ → k⊥, nk⊥ \|ki = λi Idki . We also note that in both cases below the inertia operators are positive definite under the restrictions 0 < λ1, 0 < λ2, λ1(√ 2 + 1 )2 < λ2 < λ1(√ 2− 1 )2 . Case (a2n−1, cn, a2n−2 ⊕ t). g = su(2n) = { A ∈ sl(2n,C) \|A = −AT } , g′1 = {[ Z1 Z2 −Z2 Z1 ] \|Z1 = −ZT1 , Z2 = ZT2 } ∼= sp(n), g2 = {[ Z 0 0T −t ] \|Z ∈ u(2n− 1), t = TrZ } ∼= su(2n− 1)⊕ t, k =   Z1 0 Z2 0 0T it 0T 0 −Z2 0 Z1 0 0T 0 0T −it  \|Zi ∈ gl(n− 1,C), Z1 = −ZT1 , Z2 = ZT2 , t ∈ R  ∼= sp(n− 1)⊕ t, k⊥ = X :=  Z1 v Z2 u −vT a u1 T z Z2 −u1 −Z1 v1 −uT −z −vT1 a  \|Z1 = −ZT1 , Z2 = −ZT2 , v,u,v1,u1 ∈ Cn−1, z ∈ C, a = −TrZ1  , k1 =   0 v 0 u −vT 0 uT z 0 −u 0 v −uT −z −vT 0   , k2 =   Z1 v1 Z2 0 −vT1 a uT1 0 Z2 −u1 −Z1 0 0 0 0 a  \| a = −TrZ1  , 28 K. Lompert and A. Panasyuk 1 2 ( nk⊥ + n∗k⊥ ) X =  λ2Z1 λ2v + (λ1−λ2) 2 v1 λ2Z2 λ1u + (λ1−λ2) 2 u1 −λ2v T − (λ1−λ2) 2 vT1 λ2a λ2u1 T + (λ1−λ2) 2 uT λ1z λ2Z2 −λ2u1 − (λ1−λ2) 2 u −λ2Z1 λ1v1 + (λ1−λ2) 2 v −λ1u T − (λ1−λ2) 2 uT1 −λ1z −λ1v T 1 − (λ1−λ2) 2 vT λ2a  . Case (dn+1, bn, an ⊕ t), cf. [11, solution to Exercise B.3, Chapter VI]. g = {[ W Z Z W ] \|Z,W ∈ gl(n+ 1,C), W = −W T , Z = −ZT } ∼= so(2n+ 2,R), g1 =   0 uT 0 uT −u W1 −u Z1 0 uT 0 uT −u Z1 −u W 1  \|Z1,W1 ∈ gl(n,C), Z1 = −ZT1 , W1 = −W T 1 , u ∈ Cn  ∼= so(2n+ 1,R), g2 = {[ A 0 0 −AT ] \|A ∈ u(n+ 1) } ∼= u(n+ 1), k =   0 0T 0 0T 0 W1 0 0 0 0T 0 0T 0 0 0 W 1  \|W1 = −W T 1  ∼= u(n), k⊥ = X :=  ia uT 0 vT −u 0 −v Z1 0 vT −ia uT −v Z1 −u 0  \|Z = −ZT , a ∈ R, u,v ∈ Cn  , k1 =   0 uT 0 uT −u 0 −u Z1 0 uT 0 uT −u Z1 −u 0   , k2 =   ia uT1 −u1 0 0 0 −ia uT1 −u1 0   , 1 2 ( nk⊥ + n∗k⊥ ) X =  λ2ia λ1u T + (λ1−λ2) 2 vT 0 λ1v T + (λ1−λ2) 2 uT −λ1u − (λ1−λ2) 2 v 0 −λ1v − (λ1−λ2) 2 u λ1Z1 0 λ1v T + (λ1−λ2) 2 uT −λ2ia λ1u T + (λ1−λ2) 2 vT −λ1v − (λ1−λ2) 2 u λ1Z1 −λ1u − (λ1−λ2) 2 v 0  . 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Lompert and A. Panasyuk [26] Panasyuk A., Bi-Hamiltonian structures with symmetries, Lie pencils and integrable systems, J. Phys. A: Math. Theor. 42 (2009), 165205, 20 pages. [27] Panasyuk A., Compatible Lie brackets: towards a classification, J. Lie Theory 24 (2014), 561–623. [28] Räıs M., L’indice des produits semi-directs E → ρ ×g, C. R. Acad. Sci. Paris Sér. A-B 287 (1978), A195– A197. [29] Sadetov S.T., A proof of the Mishchenko–Fomenko conjecture, Dokl. Math. 397 (2004), 751–754. [30] Thimm A., Integrable geodesic flows on homogeneous spaces, Ergodic Theory Dynamical Systems 1 (1981), 495–517. [31] Trofimov V.V., Fomenko A.T., Algebra and geometry of integrable Hamiltonian differential equations, Fak- torial, Moscow, 1995. [32] Turiel F.-J., Structures bihamiltoniennes sur le fibré cotangent, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), 1085–1088. [33] Whitney H., Bruhat F., Quelques propriétés fondamentales des ensembles analytiques-réels, Comment. Math. Helv. 33 (1959), 132–160. https://doi.org/10.1088/1751-8113/42/16/165205 https://doi.org/10.1088/1751-8113/42/16/165205 https://doi.org/10.1017/s0143385700001401 https://doi.org/10.1007/BF02565913 https://doi.org/10.1007/BF02565913 1 Introduction 2 Invariant Nijenhuis tensors on homogeneous spaces 3 Bi-Poisson structures, kroneckerity, G-invariance, and complete families of functions in involution 4 Bi-Poisson structures on cotangent bundles related to Nijenhuis (1,1)-tensors 5 Algebraic criterion of kroneckerity in the case of a locally free action 6 Applications: two homogeneous spaces with integrable geodesic flows 7 Concluding remarks A Compact real forms of the triples (g,g1,g2) and inertia operators References
id	nasplib_isofts_kiev_ua-123456789-210239
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn	1815-0659
language	English
last_indexed	2025-12-07T21:24:52Z
publishDate	2019
publisher	Інститут математики НАН України
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spelling	Lompert, K. Panasyuk, A. 2025-12-04T13:08:20Z 2019 Invariant Nijenhuis Tensors and Integrable Geodesic Flows / K. Lompert, A. Panasyuk // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 33 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 37J15; 37J35; 53D25 arXiv: 1812.04511 https://nasplib.isofts.kiev.ua/handle/123456789/210239 https://doi.org/10.3842/SIGMA.2019.056 We study invariant Nijenhuis (1,1)-tensors on a homogeneous space G/K of a reductive Lie group G from the point of view of integrability of a Hamiltonian system of differential equations with the G-invariant Hamiltonian function on the cotangent bundle T(G/K). Such a tensor induces an invariant Poisson tensor Π₁ on T(G/K), which is Poisson compatible with the canonical Poisson tensor ΠT(G/K). This Poisson pair can be reduced to the space of G-invariant functions on T(G/K) and produces a family of Poisson commuting G-invariant functions. We give, in Lie algebraic terms, necessary and sufficient conditions for the completeness of this family. As an application we prove Liouville integrability in the class of analytic integrals polynomial in momenta of the geodesic flow on two series of homogeneous spaces G/K of compact Lie groups G for two kinds of metrics: the normal metric and new classes of metrics related to decomposition of G to two subgroups G=G₁⋅G₂, where G/Gᵢ are symmetric spaces, K=G₁∩G₂. We are very grateful to anonymous referees for their useful remarks, which allowed us to essentially improve the quality of our paper in its final version. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Invariant Nijenhuis Tensors and Integrable Geodesic Flows Article published earlier
spellingShingle	Invariant Nijenhuis Tensors and Integrable Geodesic Flows Lompert, K. Panasyuk, A.
title	Invariant Nijenhuis Tensors and Integrable Geodesic Flows
title_full	Invariant Nijenhuis Tensors and Integrable Geodesic Flows
title_fullStr	Invariant Nijenhuis Tensors and Integrable Geodesic Flows
title_full_unstemmed	Invariant Nijenhuis Tensors and Integrable Geodesic Flows
title_short	Invariant Nijenhuis Tensors and Integrable Geodesic Flows
title_sort	invariant nijenhuis tensors and integrable geodesic flows
url	https://nasplib.isofts.kiev.ua/handle/123456789/210239
work_keys_str_mv	AT lompertk invariantnijenhuistensorsandintegrablegeodesicflows AT panasyuka invariantnijenhuistensorsandintegrablegeodesicflows

Invariant Nijenhuis Tensors and Integrable Geodesic Flows

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