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Umbilical-Type Surfaces in SpaceTime

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Recent Trends in Lorentzian Geometry

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

Abstract

A space-like surface S immersed in a 4-dimensional Lorentzian manifold will be said to be umbilical along a direction N normal to S if the second fundamental form along N is proportional to the first fundamental form of S. In particular, S is pseudo-umbilical if it is umbilical along the mean curvature vector field H and (totally) umbilical if it is umbilical along all possible normal directions. The possibility that the surface be umbilical along the unique normal direction orthogonal to H—“ortho-umbilical” surface—is also considered. I prove that the necessary and sufficient condition for S to be umbilical along a normal direction is that two independent Weingarten operators (and, a fortiori, all of them) commute, or equivalently that the shape tensor be diagonalizable on S. The umbilical direction is then uniquely determined. This can be seen to be equivalent to a condition relating the normal curvature and the appropriate part of the Riemann tensor of the space time. In particular, for conformally flat space-times (including Lorentz space forms) the necessary and sufficient condition is that the normal curvature vanishes. Some further consequences are analyzed, and the extension of the main results to arbitrary signatures and higher dimensions is briefly discussed.

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Notes

  1. 1.

    There is a long-standing tradition among mathematicians who study submanifolds to use the opposite convention, that is, ∇ for the inherited connection and \(\bar{\nabla }\) for the background connection. I stress this point here in the hope that this will avoid any possible confusion.

  2. 2.

    If one chooses, say, σ N to be the positive root of \(\sqrt{{\sigma }_{N }^{2}}\) then it may fail to be differentiable at points where the two eigenvalues of A N coincide, that is, at points where σ N = 0. Of course, one can always set an “initial” condition for G | x at any point on x ∈ S and then the differentiable solution for the vector field G is fixed. Nevertheless, this initial condition is arbitrary.

  3. 3.

    Notice the sign convention, which may not coincide with the preferred one for everybody.

  4. 4.

    Up to proportionality factors, this K coincides with K N if N is non-null. If N is null, then N can be chosen to be either ℓ or k, and K is − K k or \(-{K}_{\mathcal{l}}\), respectively.

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Acknowledgments

I thank Miguel Sánchez and the referees for some comments. Supported by grants FIS2010-15492 (MICINN), UFI 11/55 and GIU06/37 (UPV/EHU) and P09-FQM-4496 (J. Andalucía—FEDER).

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  1. Física Teórica, Universidad del País Vasco, Apartado 644, 48080, Bilbao, Spain

    José M. M. Senovilla

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  1. José M. M. Senovilla
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Correspondence to José M. M. Senovilla .

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  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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Senovilla, J.M.M. (2012). Umbilical-Type Surfaces in SpaceTime. 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_3

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