On the Integrability of Geodesic Flows of Submersion Metrics

@article{Jovanovic2002OnTI,
title={On the Integrability of Geodesic Flows of Submersion Metrics},
author={B. Jovanovic},
journal={Letters in Mathematical Physics},
year={2002},
volume={61},
pages={29-39}
}
• B. Jovanovic
• Published 2002
• Mathematics, Physics
• Letters in Mathematical Physics
Suppose we are given a compact Riemannian manifold (Q,g) with a completely integrable geodesic flow. Let G be a compact connected Lie group acting freely on Q by isometries. The natural question arises: will the geodesic flow on Q/G equipped with the submersion metric be integrable? Under one natural assumption, we prove that the answer is affirmative. New examples of manifolds with completely integrable geodesic flows are obtained.
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References

SHOWING 1-10 OF 21 REFERENCES
Integrable geodesic flows on the suspensions of toric automorphisms
• Mathematics
• 1999
For any toric automorphism with only real eigenvalues a Riemannian metric with an integrable geodesic flow on the suspension of this automorphism is constructed. A qualitative analysis of such a flowExpand
On the topology of manifolds with completely integrable geodesic flows
We show that if M is a compact simply connected Riemannian manifold whose geodesic flow is completely integrable with periodic integrals, then M is rationally elliptic, i.e. the rational homotopy ofExpand
Noncommutative Integrability, Moment Map and Geodesic Flows
• Mathematics, Physics
• 2001
The purpose of this paper is to discuss the relationship betweencommutative and noncommutative integrability of Hamiltonian systemsand to construct new examples of integrable geodesic flowsExpand
New examples of manifolds with completely integrable geodesic flows
• Mathematics
• 1990
Abstract We construct Riemannian manifolds with completely integrable geodesic flows, in particular various nonhomogeneous examples. The methods employed are a modification of Thimm′s method,Expand
Integrable geodesic flows on homogeneous spaces
• Mathematics
• 2001
It is proved that the geodesic flow of a bi-invariant metric on an arbitrary homogeneous space of a compact Lie group is Liouville-integrable in the non-commutative sense.
Integrable geodesic flows with positive topological entropy
• Mathematics, Physics
• 2000
An example of a real-analytic metric on a compact manifold whose geodesic flow is Liouville integrable by $C^\infty$ functions and has positive topological entropy is constructed.
New examples of manifolds with strictly positive curvature
Berger [-3], Wallach [10] and Berard Bergery [2] have classified all simply connected smooth manifolds which allow a homogeneous Riemannian metric of strictly positive curvature. Besides the rank oneExpand
Generalized Liouville method of integration of Hamiltonian systems
• Mathematics
• 1978
In this paper we shall show that the equations of motion of a solid, and also Liouville's method of integration of Hamiltonian systems, appear in a natural manner when we study the geometry of levelExpand
Theory of Tensor Invariants of Integrable Hamiltonian Systems. II. Theorem on Symmetries and Its Applications
Abstract: The theorem on symmetries is proved that states that a Liouville-integrable Hamiltonian system is non-degene\-rate in Kolmogorov's sense and has compact invariant submanifolds if and onlyExpand
Annals of global analysis and geometry
• Mathematics
• 1983
▶ Examines global problems of geometry and analysis ▶ Looks at interactions between differential geometry and global analysis and their application to problems of theoretical physics ▶ Contributes toExpand