About every convex set in any generic Riemannian manifold

TL;DR

This paper establishes necessary conditions for geodesics on convex hypersurfaces in Riemannian manifolds and derives universal properties of convex sets, such as strict convexity when bounded by smooth hypersurfaces.

Contribution

It introduces a necessary condition for geodesics on convex hypersurfaces and reveals that convex sets bounded by smooth hypersurfaces are strictly convex in any generic Riemannian manifold.

Findings

Geodesics on convex hypersurfaces satisfy specific necessary conditions.

Convex sets bounded by smooth hypersurfaces are strictly convex.

Universal properties of convex sets hold in generic Riemannian manifolds.

Abstract

We give a necessary condition on a geodesic in a Riemannian manifold that can run in some convex hypersurface. As a corollary we obtain peculiar properties that hold true for every convex set in any generic Riemannian manifold (M,g). For example, if a convex set in (M,g) is bounded by a smooth hypersurface, then it is strictly convex.