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Infinitesimal and Local Convexity of a Hypersurface in a Semi-Riemannian Manifold

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Recent Trends in Lorentzian Geometry

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 26))

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. [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.

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Notes

  1. 1.

    According to the case when H is the graph of a function and n is the smooth unit vector which lies on the same side of the graph as the canonic unit vector defining the axis of the values of the function, from [9], we know that the geodesics issuing from p 0 and tangent to H are locally contained in the closure of the component of a tubular neighborhood of H individuated by − n. As the sign of Π changes if one changes n with − n, it is clear that what is really important in this definition is the fact that Π is semidefinite in U, either positive or negative. Clearly in the case when Π is negative semidefinite the geodesics tangent at p 0 to H are locally contained in the component individuated by n itself.

  2. 2.

    In some cases, the Chern connection which is a linear connection on the vector bundle π∗ TM over \(TM \setminus \{ 0\}\), \(\pi : TM \rightarrow M\) the natural projection, reduces to a linear connection on M, even if the Finsler metric is not a Riemannian one. When this happens the Finsler metric is said of Berwald type. Since from a theorem of Szabó (cf. [1, Sect. 10.1]), given a Berwald metric F on M, there exists a Riemannian metric such that its Levi-Civita connection coincides with the Chern connection of F, we can state, as already observed in [10], that Bishop’s proof is also valid in any Berwald space.

  3. 3.

    Observe that such geodesics must be causal and future pointing as the points \({\gamma }_{k}({s}_{kj})\) and \({\gamma }_{k}\left({s}_{k(j+1)}\right)\) are causally related, see [27, Chap. 14, Lemma 2].

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Acknowledgements

I would like to thank M. Sánchez for suggesting to investigate the topic of this note and for several useful comments. Moreover, I thank the local organizing committee of the “VI International Meeting on Lorentzian Geometry, Granada 2011” for the financial support and the hospitality during the workshop.

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Authors and Affiliations

  1. Dipartimento di Meccanica, Matematica e Managment, Politecnico di Bari, via Orabona 4, 70125, Bari, Italy

    Erasmo Caponio

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  1. Erasmo Caponio
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Correspondence to Erasmo Caponio .

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Editors and Affiliations

  1. , Dept of Geometry and Topology, University of Granada, Campus Fuentenueva s/n, Granada, E18071, Granada, Spain

    Miguel Sánchez

  2. Departamento de Mátematica Aplicada, Universidad de Granada, Campus de Fuentenueva, Granada, 18071, Spain

    MIguel Ortega

  3. , Dept. of Geometry and Topology, University of Granada, Campus Fuentenueva s/n, Granada, E-18071, Granada, Spain

    Alfonso Romero

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Caponio, E. (2012). Infinitesimal and Local Convexity of a Hypersurface in a Semi-Riemannian Manifold. In: Sánchez, M., Ortega, M., Romero, A. (eds) Recent Trends in Lorentzian Geometry. Springer Proceedings in Mathematics & Statistics, vol 26. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4897-6_6

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