Convexity of domains of Riemannian manifolds

TL;DR

This paper investigates the geodesic connectedness and convexity of open domains in Riemannian manifolds using variational methods, providing new insights and examples, with applications to dynamical systems.

Contribution

It proves geodesic connectedness for open domains with non-smooth, non-convex boundaries and explores conditions for convexity, including practical examples and an application to dynamical systems.

Findings

Open domains can be geodesically connected despite non-smooth, non-convex boundaries.

Conditions for convexity of domains are identified and demonstrated.

Application to existence of fixed-energy trajectories in dynamical systems.

Abstract

In this paper we analyze the problem of the geodesic connectedness of subsets of Riemannian manifolds. By using variational methods, the geodesic connectedness of open domains (whose boundaries can be not differentiable and not convex) of a smooth Riemannian manifold is proved. In some cases also the convexity of the domain is obtained. Moreover we present examples of the applicability and of the independence of the assumptions. Finally we give an application to the existence of trajectories with fixed energy of dynamical systems.