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 . We show in this note that the techniques in , 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.
|Autori interni:||CAPONIO, Erasmo|
|Titolo:||Infinitesimal and local convexity of a hypersurface in a semi-Riemannian manifold|
|Titolo del libro:||Recent Trends in Lorentzian Geometry|
|Nome editore:||Springer Science+Business Media, LLC 2011|
|Data di pubblicazione:||2013|
|Digital Object Identifier (DOI):||10.1007/978-1-4614-4897-6_6|
|Appare nelle tipologie:||2.1 Contributo in volume (Capitolo o Saggio)|