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1.
Homogeneous spaces of all compact Lie groups admit Riemannian metrics with completely integrable geodesic flows by means of C –smooth integrals [9, 10]. The purpose of this paper is to give some constructions of complete involutive algebras of analytic functions, polynomial in velocities, on the (co)tangent bundles of homogeneous spaces of compact Lie groups. This allows us to obtain new integrable Riemannian and sub-Riemannian geodesic flows on various homogeneous spaces, such as Stiefel manifolds, flag manifolds and orbits of the adjoint actions of compact Lie groups. Mathematics Subject Classification (2000): 70H06, 37J35, 53D17, 53D25  相似文献   

2.
In this paper, the differential geometry of second canonical extension2 M of a differentiable manifoldM is studied. Some vector fields tangent to2 M inTTM are determined. In addition we obtain that the second canonical extensions ofM and a totally geodesic submanifold inM are totally geodesic submanifolds inTTM and2 M respectively.  相似文献   

3.
On Lie group manifolds, we consider right-invariant magnetic geodesic flows associated with 2-cocycles of the corresponding Lie algebras. We investigate the algebra of the integrals of motion of magnetic geodesic flows and also formulate a necessary and sufficient condition for their integrability in quadratures, giving the canonical forms of 2-cocycles for all four-dimensional Lie algebras and selecting integrable cases. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 2, pp. 189–206, August, 2008.  相似文献   

4.
In this article, by modifying the argument shift method, we prove Liouville integrability of geodesic flows of normal metrics (invariant Einstein metrics) on the Ledger–Obata n-symmetric spaces K n /diag(K), where K is a semisimple (respectively, simple) compact Lie group.  相似文献   

5.
6.
In this paper we give a complete classification of simply connected homogeneous almost α-Kenmotsu three-manifolds M whose Ricci operator is invariant along the Reeb flow. We get this classification by using the Gaussian and the extrinsic curvature associated with the canonical foliation of M.  相似文献   

7.

In this paper, we study the invariant metrizability and projective metrizability problems for the special case of the geodesic spray associated to the canonical connection of a Lie group. We prove that such canonical spray is projectively Finsler metrizable if and only if it is Riemann metrizable. This result means that this structure is rigid in the sense that considering left invariant metrics, the potentially much larger class of projective Finsler metrizable canonical sprays, corresponding to Lie groups, coincides with the class of Riemann metrizable canonical sprays. Generalisation of these results for geodesic orbit spaces are given.

  相似文献   

8.
We show that a C1 torus that is homologous to the zero section, invariant by the geodesic flow of a symmetric Finsler metric in T2, and possesses closed orbits is a graph of the canonical projection. This result, together with the result obtained by Bialy in 1989 for continuous invariant tori without closed orbits of symmetric Finsler metrics in T2, shows that the second Birkhoff Theorem holds for C1 Lagrangian invariant tori of symmetric Finsler metrics in the two torus. We also study the first Birkhoff Theorem for continuous invariant tori of Finsler metrics in T2 and give some sufficient conditions for a continuous minimizing torus with closed orbits to be a graph of the canonical projection. Partially supported by CNPq, FAPERJ, TWAS  相似文献   

9.
Summary We construct definitely the automorphism group of a Sasakian space form ¯M=E 2m+1 (–3) and study the existence of a totally geodesic invariant submanifold of ¯M tangent to a given invariant subspace in the tangent space of ¯M. We also study the Frenet curves in ¯M under a totally contact geodesic immersion of a contact CR-submanifold into ¯M. The purpose of this paper is to prove a reduction theorem of the codimension for a totally contact geodesic, contact CR-submanifold of ¯M.  相似文献   

10.
We propose a new condition à{{\aleph}} which enables us to get new results on integrable geodesic flows on closed surfaces. This paper has two parts. In the first, we strengthen Kozlov’s theorem on non-integrability on surfaces of higher genus. In the second, we study integrable geodesic flows on a 2-torus. Our main result for a 2-torus describes the phase portraits of integrable flows. We prove that they are essentially standard outside what we call separatrix chains. The complement to the union of the separatrix chains is C 0-foliated by invariant sections of the bundle.  相似文献   

11.
We consider the sequence of the hyperspheres M n , i.e., the homogeneous transitive spaces of the Cartan subgroup of the group and study the normalized limit of the corresponding sequence of invariant measures m n on those spaces. In the case of compact groups and homogeneous spaces, for example, for the classical pairs (SO(n), S n-1), n = 1, 2, … , the limit of the corresponding measures is the classical infinite-dimensional Gaussian measure; this is the well-known Maxwell-Poincaré lemma. Simultaneously the Gaussian measure is a unique (up to a scalar) invariant measure with respect to the action of the infinite orthogonal group O(∞). This coincidence implies the asymptotic equivalence between grand and small canonical ensembles for the series of the pairs (SO(n), S n-1). Our main result shows that the situation for noncompact groups, for example for the case , is completely different: the limit of the measures m n does not exist in the literal sense, and we show that only a normalized logarithmic limit of the Laplace transforms of those measures does exist. At the same time, there exists a measure which is invariant with respect to a continuous analogue of the Cartan subgroup of the group GL(∞), the so-called infinite-dimensional Lebesgue measure (see [7]). This difference is an evidence for non-equivalence between the grand and small canonical ensembles in the noncompact case. To my friend Dima Arnold  相似文献   

12.
The “Projective Rank” of a compact connected irreducible Hermitian symmetric space M has been defined as the maximal complex dimension of the compact totally geodesic complex submanifolds having positive holomorphic bisectional curvature with the induced K?hler metric. We present a geometric way to compute this invariant for the space M based on ideas developed in [1], [13] and [14]. As a consequence we obtain the following inequality relating the Projective Rank, Pr(M), the usual rank,rk(M), and the 2-number # (which is known to be equal to the Euler-Poincare characteristic in these spaces). Received: 6 June 2000 / Published online: 1 February 2002  相似文献   

13.
We prove that the classical integrability condition for almost complex structures on finite-dimensional smooth manifolds also works in infinite dimensions in the case of almost complex structures that are real analytic on real analytic Banach manifolds. With this result at hand, we extend some known results concerning existence of invariant complex structures on homogeneous spaces of Banach–Lie groups. By way of illustration, we construct the complex flag manifolds associated with unital C*-algebras.Mathematics Subject Classifications (2000): primary 32Q60; secondary 53C15, 58B12.  相似文献   

14.
A Riemannian manifold M is called 2-flat homogeneous if every geodesic is contained in some 2-flat , and if the group of isometries of M acts transitively on the set of pairs (p, ) with p . By a 2-flat we mean a closed, connected, flat, totally geodesic, 2-dimensional submanifold of M. It is proved in the paper that 2-flat homogeneous spaces are symmetric.  相似文献   

15.
We prove that solutions for ¯ get 1/M-derivatives more than the data in Lp-Sobolev spaces on a bounded convex domain of finite type M by means of the integral kernel method. Also we prove that the Bergman projection is invariant under the Lp-Sobolev spaces of fractional orders by different methods from McNeal-Stein's. By using these results, we can get Lp-Sobolev estimates of order 1/M for the canonical solution for ¯. The author was supported by grant No. R01-2000-000-00001-0 from the Basic Research Program of the Korea Science&Engineering Foundation.  相似文献   

16.
The “Projective Rank” of a compact connected irreducible Hermitian symmetric space M has been defined as the maximal complex dimension of the compact totally geodesic complex submanifolds having positive holomorphic bisectional curvature with the induced K?hler metric. We present a geometric way to compute this invariant for the space M based on ideas developed in [1], [13] and [14]. As a consequence we obtain the following inequality relating the Projective Rank, the usual rank, and the 2-number (which is known to be equal to the Euler-Poincare characteristic in these spaces). Received: 6 June 2000 / Revised version: 6 August 2001 / Published online: 4 April 2002  相似文献   

17.
Let Mn be a closed Riemannian manifold with a nontrivial second homology group. In this paper we prove that there exists a geodesic net on Mn of length at most 3 diameter(Mn). Moreover, this geodesic net is either a closed geodesic, consists of two geodesic loops emanating from the same point, or consists of three geodesic segments between the same endpoints. Geodesic nets can be viewed as the critical points of the length functional on the space of graphs immersed into a Riemannian manifold. One can also consider other natural functionals on the same space, in particular, the maximal length of an edge. We prove that either there exists a closed geodesic of length ≤ 2 diameter(Mn), or there exists a critical point of this functional on the space of immersed θ-graphs such that the value of the functional does not exceed the diameter of Mn. If n=2, then this critical θ-graph is not only immersed but embedded.Mathematics Subject Classifications (2000). 53C23, 49Q10  相似文献   

18.
The purpose of this paper is to discuss the relationship betweencommutative and noncommutative integrability of Hamiltonian systemsand to construct new examples of integrable geodesic flows onRiemannian manifolds. In particular, we prove that the geodesic flowof the bi-invariant metric on any bi-quotient of a compact Lie group isintegrable in the noncommutative sense by means of polynomial integrals, andtherefore, in the classical commutative sense by means ofC -smooth integrals.  相似文献   

19.
By using commutator methods, we show uniform resolvent estimates and obtain globally smooth operators for self-adjoint injective homogeneous operators H on graded groups, including Rockland operators, sublaplacians, and many others. Left or right invariance is not required. Typically the globally smooth operator has the form T = V|H|1∕2, where V only depends on the homogeneous structure of the group through Sobolev spaces, the homogeneous dimension and the minimal and maximal dilation weights. For stratified groups improvements are obtained, by using a Hardy-type inequality. Some of the results involve refined estimates in terms of real interpolation spaces and are valid in an abstract setting. Even for the commutative group ?N some new classes of partial differential operators are treated.  相似文献   

20.
Adrien Deloro 《代数通讯》2013,41(5):1981-2008
We identify the spaces of homogeneous polynomials in two variables 𝕂[Yk, XYk?1, ?, Xk] among representations of the Lie ring 𝔰𝔩2(𝕂). This amounts to constructing a compatible 𝕂-linear structure on some abstract 𝔰𝔩2(𝕂)-modules, where 𝔰𝔩2(𝕂) is viewed as a Lie ring.  相似文献   

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