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.

Infinitesimal and local convexity of a hypersurface in a semi-Riemannian manifold / Caponio, Erasmo - In: Recent Trends in Lorentzian Geometry / [a cura di] Miguel Sánchez, Miguel Ortega, Alfonso Romero. - STAMPA. - New York, NY : Springer, 2012. - ISBN 978-1-4614-4896-9. - pp. 163-177 [10.1007/978-1-4614-4897-6_6]

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

Caponio, Erasmo
2012-01-01

Abstract

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.
2012
Recent Trends in Lorentzian Geometry
978-1-4614-4896-9
Springer
Infinitesimal and local convexity of a hypersurface in a semi-Riemannian manifold / Caponio, Erasmo - In: Recent Trends in Lorentzian Geometry / [a cura di] Miguel Sánchez, Miguel Ortega, Alfonso Romero. - STAMPA. - New York, NY : Springer, 2012. - ISBN 978-1-4614-4896-9. - pp. 163-177 [10.1007/978-1-4614-4897-6_6]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11589/11804
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