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

Given a Riemannian manifold (M, g) and an embedded hypersurface H in M, a result by R. L. Bishop states that infinitesimal convexity on a neighborhood of a point in H implies local convexity. Such result was extended very recently to Finsler manifolds by the author et al. in [2]. We show in this note that the techniques in [2], unlike the ones in Bishop’s paper, can be used to prove the same result when (M, g) is semi-Riemannian. We make some remarks for the case when only time-like, null, or space-like 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.