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

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.

14 pages, AMSLaTex, 2 figures. v2: typos fixed, added one reference and several comments, statement of last proposition corrected

References in corpus (3)

• On the energy functional on Finsler manifolds and applications to stationary spacetimes
• On the interplay between Lorentzian Causality and Finsler metrics of Randers type
• Convex domains of Finsler and Riemannian manifolds

Cited by in corpus (2)

• Convex regions of stationary spacetimes and Randers spaces. Applications to lensing and asymptotic flatness
• Lorentzian-Euclidean black holes and Lorentzian to Riemannian metric transitions