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Singularities of integrable geodesic flows on multidimensional torus and sphere. (English) Zbl 0849.58053

The author investigates geodesic flows of Liouville metrics on the torus \(T^n\) and of standard metrics on the sphere \(S^n\) (and their perturbations). These flows are integrable and it is assumed that first integrals are independent almost everywhere and their common level sets are compact. Thus, an associated singular foliation by Liouville tori appears. A germ of the foliation near a singular leaf is called a nonlocal singularity.
The paper presents a topological description of nonlocal singularities for the considered geodesic flows.

MSC:

37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
53C22 Geodesics in global differential geometry
58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory
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