On periodic orbits in cotangent bundles of non-compact manifolds

TL;DR

This paper extends the existence results of periodic orbits on energy hypersurfaces in cotangent bundles from Euclidean spaces to more general Riemannian manifolds with flat ends, under certain geometric conditions.

Contribution

It generalizes previous results by proving the existence of periodic orbits on a broader class of hypersurfaces in cotangent bundles of non-compact Riemannian manifolds.

Findings

Periodic orbits exist on certain energy hypersurfaces in cotangent bundles of Riemannian manifolds with flat ends.

Extension of previous Euclidean space results to non-compact manifolds.

Conditions involving homology groups influence the existence of periodic orbits.

Abstract

This paper is concerned with the existence of periodic orbits on energy hypersurfaces in cotangent bundles of Riemannian manifolds defined by mechanical Hamiltonians. In \cite{bpv} it was proved that, provided certain geometric assumptions are satisfied, regular mechanical hypersurfaces in $\mathbb{R}^{2n}$, in particular non-compact ones, contain periodic orbits if one homology group among the top half does not vanish. In the present paper we extend the above mentioned existence result to a class of hypersurfaces in cotangent bundles of Riemannian manifolds with flat ends.