Infinitesimal and local convexity of a hypersurface in a semi-Riemannian manifold

TL;DR

This paper extends the equivalence of infinitesimal and local convexity from Riemannian to semi-Riemannian manifolds, with applications to geodesic connectedness in Lorentzian geometry.

Contribution

It demonstrates that infinitesimal convexity implies local convexity in semi-Riemannian manifolds, generalizing a known Riemannian result, and discusses implications for geodesic connectedness.

Findings

Infinitesimal convexity implies local convexity in semi-Riemannian manifolds.

Remarks on convexity involving timelike, null, or spacelike geodesics.

Applications to geodesic connectedness in Lorentzian manifolds.

Abstract

Given a Riemannian manifold M and a hypersurface H in M, it is well known that infinitesimal convexity on a neighborhood of a point in H implies local convexity. We show in this note that the same result holds in a semi-Riemannian manifold. We make some remarks for the case when only timelike, null or spacelike geodesics are involved. The notion of geometric convexity is also reviewed and some applications to geodesic connectedness of an open subset of a Lorentzian manifold are given.