Vladimir S. Matveev (Jena) Generalisation of Benenti systems for Kähler manifolds.

Transcription

1 Vladimir S. Matveev (Jena) Generalisation of Benenti systems for Kähler manifolds.

2 What is a Benenti system? Benenti system on M n is a pair (g,l), where g is a (pseudo)-riemannian metric, and L a symmetric (0,2) tensor satisfying L ij,k = g ik λ j + g jk λ i ( ) for a certain covector λ = λ i (actually, immediately λ i = 1 2 (L pqg pq ), so the equation ( ) is equation on L only). Trivial example: L = g; λ i = 0. Different notions of non-triviality: Matveev-Topalov: L const g. Sinjukov-Mikes-Kiosak: λi = 0 ( L covariantly constant). Benenti: All eigenvalues of L i j are nonconstant (if g this Riemannian, this imply that L i j has n = dim(m) different eigenvalues. )

3 Why people studied Benenti systems? Argument A. They appear naturally in the theory of projectively equivalent metrics and in other geometric problems. Def. Two metrics g and ḡ on the same manifold are projectively (or geodesically) equivalent, if every g-geodesic (considered as a unparameterized curve) is a ḡ-geodesic. Theorem (Sinjukov 1966). g and ḡ are projectively equivalent, if ( 1/(n+1) and only if (g, L) with L ij := (g pi ḡ pq g qj ) det(ḡ) det(g)) is a Benenti structure. Historical remark. Theory of geodesically equivalent metrics is much older than this result of Sinjukov the first examples are due to Lagrange 1789; in 19th century, Beltrami, Dini and Levi-Civita already obtained nontrivial results. Nevertheless, all resent progress in the theory of geodesically equivalent metrics essentially uses the resulf of Sinjukov.

4 Why people studied Benenti systems? Argument B. They appear naturally is the theory of integrable systems. Theorem (Painleve 1896; Topalov-Matveev 1997) If (g,l) is a Benenti systems, then for every t 1,t 2 R the quadratic in velocities functions I t : TM R, I t (ξ) = g(det(l t id) (L t id) 1 ξ,ξ) are commutative integrals for the geodesic flow of g. (Here we understand L as an (1,1)-tensor, i.e., as a linear operator on TM). In the case when L has n different eigenvalues, the integrals are functionally independent and the geodesic flow is Liouville-integrable. Many physically interesting integrable systems can be obtained by this form (a series of works of Benenti ) There is no problem to introduce potential energy in the picture (Benenti, Kruglikov-Matveev 2006) One can also generalize the integrability to the quantum version of the geodesic flows

5 The integrals for Benenti systems are easy to handle 1. They are quadratic in velocities 2. the corresponding quadratic forms are simultaneously diagonalizable 3. which implies that (when all eigenvalues of L are not constant) that the variables in the Hamilton-Jacobi equation can be separated. 4. In dimension two, every integral quadratic in velocities can be obtained with the help of Benenti system (essentially Darboux; see also Bolsinov-Matveev-Pucacco). In dimension three, for the metric of constant positive curvature, every system of integrals satisfying (1), (2) comes from a Benenti system (Schöbel 2011)

6 Why people studied Benenti systems? Argument C: Benenti systems appear independently in different parts of mathematics Benenti systems were studied by Benenti Bolsinov Braden Ibort Magri Marmo Crampin Sarlet Tondo Topalov Saunders Cantrijn Kolokoltsev Rastelli Chanu Marciniak Ranada Santander Kiyohara Smirnov Horwood Bolsinov Fomenko Kozlov under the names L systems, BM-systems, Benenti-systems, cofactor systems, quasi-bi-hamiltonian systems, systems admitting special conformal Killing tensor and appeared many times during SPT-conferences

7 Why people studied Benenti systems? Argument D. The class of Benenti systems is relatively big and provides us with a lot of interesting examples. Theorem (Dini 1869): In dimension 2, every nontrivial Benenti system is given in a local coordinate system in a neighborhood of almost every point by g = (X(x) Y (y))(dx 2 + dy 2 ) L i j = diag(x(x),y(y)). A similar result holds in all dimensions (Levi-Civita 1896); in all dimensions there is a functional freedom in choosing a Benenti system

8 Is there any sense to study Benenti systems in the Kähler situation? Kähler manifold = (M 2n,g,J), were the (1,1)-tensor J is covariantly constant, is g-selfadjoint, and satisfies J 2 = id. Theorem (T. Otsuki, Y. Tashiro 1954; for Riemannian metrics follows from Levi-Civita 1896): Suppose L is Benenti structure on Kähler (M 2n,g,J). Assume in addition that L is complex, that is, L(J,J ) = L(, ). Then, L is covariantly constant. Thus, if L is complex, then the Benenti system is somehow trivial. Theorem (Folklore) Suppose L is Benenti structure on Kähler (M 2n,g,J). Assume L is not covariantly constant and g is not flat. Then, there exists a linear combination const 1 L + const 2 g such that it has rank at most one at every point. Moreover, the metric g is the metric of the flat cone over a Sasaki manifold: locally, g = dx (x 1) 2 h αβ, where h αβ (α,β = 2,...,2n) is a Sasaki metric on a (2n 1)-dimensional manifold with coordinates x 2,...,x 2n. Thus, there is no sense to study Benenti systems on Kähler manifolds

9 An analog of the Benenti system for Kähler manifold On Kähler M 2n, we will consider a symmetric (0,2) tensor satisfying L ij,k = λ i g jk + λ j g ik λ p J p i J jk λ p J p j J ik. ( ) for a certain covector λ = λ i (actually, immediately λ i = 1 4 (Lp p) = trace g (L), so the equation ( ) is equation on L only). Main message of the talk: L satisfying ( ) has many properties similar to that of Benenti structure, and can be studied by similar methods. New intersting phemomena appear. All in one: it is worse to study it.

10 Similarity with Benenti systems Argument A. Equation ( ) appears naturally is the theory of h-projectively equivalent metrics and in other geometric problems. Def. A regular curve γ : I M is called h-planar with respect to g if for some functions α(t), β(t). γ(t) γ = α(t) γ(t) + β(t)j γ(t) t I infinitely many h-planar curves γ with γ(0) = x and γ(0) = ζ for each x M and ζ T x M. x ζ γ reparameterized geodesics satisfy γ γ = α γ. γ γ = (α + iβ) γ complex geodesics

11 h projective equivalence is equivalent to ( ) Def. Kähler metrics g,ḡ on (M,J) are h-projectively equivalent h-planar curves of g and ḡ coincide. Let g,ḡ be Kähler metrics on (M,J). Consider g-self-adjoint, complex, non-degenerate (0,2)-tensor L ij = ( det ḡ det g ) 1 2(n+1) ḡ pq g ip g qj Theorem (Mikes, Domashev 1978). ḡ is h-projectively equivalent to g L ij satisfies ( ) Historical remark. h projectively equivalent metrics were introduced in T. Otsuki, Y. Tashiro Their motivation was quite naive: they realized that projectively equivalent metrics are not interesting in the Kähler situation, and looked for a complex analog of the projective equivalence. During , theory of h-projectively equivalent metrics was one of the main research directions in Japanese and Soviet (Odessa, Kazan) differential geometry schools. Independently, an algebraically equivalent object was introduced in Apostolov-Calderbank-Gaudushon 2004 under the name hamiltonian 2-forms, see publications of V. Apostolov, D. Calderbank, P. Gauduchon, J. Diff. Geom, Investiones, Comm. Anal. Geom in ).

12 Argument B. They appear naturally in the theory of integrable systems Theorem (Topalov 2003) The functions F t : TM R, t R, I t (ζ) = g( det (L tid) (L tid) 1 ζ,ζ) are a family of commuting integrals for the geodesic flow of g. We have 2n dimensional manifold; the family I t is at most n dimensional. Theorem (Apostolov et al 2004/ Topalov-Kiyohara 2010/ Matveev-Rosemann 2010) The functions K t : TM R, K t (ξ) = ξ p J i p det(l t id) x i are a family of commuting integrals for the geodesic flow of g. The integrals ( are linear, the corresponding Killing vector field is det(l ) sgrad t id). Theorem All integrals above commute. In the case when all eigenvalues of L are not constant, we obtain 2n-functionally independent implying the geodesic flow is Liouville integrable.

13 Integrals are interesting and provide with new phenomena 1. They are quadratic (or even linear) in velocities 2. The quadratic forms corresponding to I t are simultaneously diagonalizable. The integrals K t are also simultaneously diagonalizable. But all integrals together are NOT simultaneously diagonalizable. This phenomena is (up to my knowledge) new; it is the first big family of integrable systems with this property. Natural questions; the answer for Benenti systems were obtained by standard participants of SPT-conferences. ( ) Are there physically interesting integrable systems of this form? ( ) How to introduce potential energy in the picture? ( ) Does the quantum version of the geodesic flow is quantum-integrable? (In dimension 4, the answer is positive straightforward calculations).

14 Argument C: Appeared independently in different branches of mathematics and were studies by different group of mathods Classical differential geometry. T. Otsuki, Y. Tashiro, K. Yano, M. Obata, S. Tanno, H. Akbar-Zadeh,... (1950th 1980th); Sinjukov, Mikes, Aminova, Kiosak (1980th 2000). Methods: tensor calculus and Bochner trick. Integrable systems: K. Kiyohara, M. Igarashi, P. Topalov. Method: PDE-analysis. Kähler geometry: V. Apostolov, D. Calderbank, P. Gauduchon, C. Tønnesen-Friedman. Methods: Symplectic and complex machinery.

15 Argument D. The class of (g, J, L) satisfying ( ) is relatively big and provides us with a lot of interesting examples. Recall that local classification of projectively equivalent Riemannian metrics is due to Levi-Civita 1896; there is a functional freedom in choosing (g,l). Local classification of h projectively equivalent metrics was obtain in the most interesting partial case when all eigenvalues of L are not constant follows from Kiyohara 1997 and Topalov (See next slide). In the general case, the local classification is due to Apostolov-Calderbank-Gauduchon Locally, (g,j,l) depend on functional parameters. The last papers of Apostolov et al (Invent. Math and Comm. Anal. Geom. 2008) were dedicated to construction of examples interesting for Kähler geometry. Question. Are there interesting examples for mathematical physics and for dynamical systems?

16 Local description of in 4D (under nondegeneracy assumptions) (Topalov;Igarashi-Kiyohara; Apostolov et al). L ij = ¾ g = ¾ x 1 x 2 X(x 1 ) 0 x 1 x 2 Y(x 2 ) X(x 1)+Y(x 2 ) x 1 x 2 X(x 1)x 2 +Y(x 2 )x 1 x 1 x X(x 1)x 2 +Y(x 2 )x 1 x 1 x 2 X(x 1)x 2 2 +Y(x2 )x 1 2 x 1 x 2 J i j = 2 (x 1 x 2)x 1 X(x 1 ) ¾ x 1 X(x 1 ) x 2 Y(x 2 ) X(x 1 ) x 1 x 2 X(x 1 )x 2 x 1 x 2 Y(x 2 ) x 2 x 1 Y(x 2 )x 1 x 2 x (X (x 1 )) 1 (Y (x 2 )) x 2(x 2 x 1 ) Y(x 2 ), x 1 X(x 1) x 2 Y(x 2 ) x 1 x 2 2 x 1 x 2(X(x 1 ) Y(x 2 )) x 1 x 2, x 1 x 2(X(x 1 ) Y(x 2 )) x 1 x 2 2 x 1 x 2(X(x 1 )x 2 Y(x 2 )x 1 ) x 1 x 2

17 Two main messages of my talk. This object is new and was not really studies in the context of mathematical physics. Do it!!! The object was studied by different groups/methods, and people from one group did not know about the existence of other group. Combine the methods!!! I will show one (successful) application.

18 (Successful) application: h-projective transformations and Yano-Obata conjecture Let (M, g, J) be a Kähler manifold. Def. A diffeomorphism f : M M is called h-projective transformation if f is holomorphic (that is, f J = J) and f sends h-planar curves to h-planar curves. We denote by HPro 0 (g, J) the connected component containing the identity of the Lie-group of h-projective transformations. Clearly, Iso 0 (g, J) HPro 0 (g, J). Theorem ( Yano-Obata conjecture, Matveev Rosemann, arxiv: v1, 2011). Let (M, g, J) be a compact, connected Riemannian Kähler manifold of real dimension 2n 4. If (M, g, J) cannot be covered by (CP(n), c g FS, J standard ) for some c > 0, then Iso 0 = HPro 0. (Here g FS is the Fubini-Studi metric).

19 Are the assumptions in the theorem necessary? Theorem 1 ( Yano-Obata conjecture ) Let (M,g,J) be a compact, connected Riemannian Kähler manifold of real dimension 2n 4. If (M,g,J) cannot be covered by (CP(n),c g FS,J standard ) for some c > 0, then Iso 0 = HPro 0. Locally, the statement is wrong (counterexamples are given in Mikes (1998)). The compacteness condition might not be necessary: We conject that the theorem is true for complete metrics. The pseudo-riemannian Kähler case is an open problem. We do not know whether the theorem is true for metrics with arbitrary signature.

20 The assumptions in the theorem are necessary h-projective transformations on CP(n). We first describe the h-planar curves of (CP(n), g FS, J standard ): Recall: projective line L is projection of complex plane E C n+1 onto CP(n). properties of projective line L CP(n): (a) totally geodesic submanifold (b) real dimension = 2 (c) Complex (i.e., J standard (TL) = TL) For each curve γ : I L CP(n), we have γ(t) γ (a) T γ(t) L (b),(c) = span R { γ(t), J standard ( γ(t))} γ γ = α γ + βj standard ( γ) We obtain: A curve γ is h-planar with respect to g FS γ is contained in a projective line L

21 A Gl(n + 1, C) induces a bi-holomorphic transformation f A : CP(n) CP(n) by f A (span C (x)) = span C (Ax). f A maps the h-planar curves of g FS onto itself. Indeed: γ h-planar with respect to gfs γ contained in projective line π(e) (E C n+1 plane, π : C n+1 \ {0} CP(n) projection) A Gl(n + 1, C) sends complex planes E to complex planes. f A HPro(g FS, J standard ) for all A Gl(n + 1, C) f A Iso(g FS, J standard ) A is proportional to a unitary matrix. Iso 0 (g FS, J standard ) HPro 0 (g FS, J standard ) Yano-Obata conjecture: (CP(n), g FS, J standard ) is the only (up to covering and multiplication of the metric with a constant) compact, connected Riemannian Kähler manifold with Iso 0 HPro 0.

22 Special cases of the Yano-Obata conjecture were proven before Japan (Obata, Yano) France (Lichnerowicz) UdSSR (Sinjukov) Yano, Hiramatu 1981: Akbar-Zadeh 1988: Mikes 1978: constant scalar curvature Ricci-flat locally symmetric

23 Conjectures of this type were standard in the Projective Obata-Lichnerowicz-Conjecture (proved in Matveev 2007): (M n, g) complete, connected Riemannian manifold which cannot be covered by (S n, c g Standard ). Then Pro 0 (g) = Iso 0 (g). Conformal Obata-Lichnerowicz-Conjecture (proved in Obata 1971 compact, Alekseevskii 1972, Ferrand 1994, Schoen 1995): (M n, g) connected Riemannian manifold which is not conformally equivalent to S n or E n. Then C 0 (g) = Iso 0 (σ g) after a conformal change of the metric.

24 Main equation and degree of mobility L ij,k = λ i g jk + λ j g ik λ p J p i J jk λ p J p j J ik. ( ) where λ = 1 4 trace g(l). Thus is a linear system of PDEs on the (0, 2)-tensor L. The space of solutions is a linear vector space. Def. Degree of mobility D(g) of Kähler metric g = is the dimension of space of complex, self-adjoint solutions L of the main equation. It is known (Otsuki, Y. Tashiro 1954/Mikes-Domashev 1978) that 1 D(g) <

25 Plan of the proof of the theorem If D(g) = 1, every metric ḡ that is h-projectively equivalent to g, is already proportional to g, ḡ = const g. In particular, every h-projective transformation f is a homothety, f g = const g. On compact manifolds, f is already an isometry. D(g) > 2 was done in Fedorova, Kiosak, Matveev, Rosemann arxiv: , 2010: the proof involves prolongation-projection method, theory of Frobenius systems, nontrivial tensor calculus, uses a nontrivial result of Tanno 1978, and will not be explained here The proof of the remaining case D = 2 is done in arxiv: v1; it essentially uses integrable systems.