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Hyperbolic Metrics on Riemann Surfaces and Space-Like CMC-1 Surfaces in de Sitter 3-Space Read chapter Read first chapter

2013 | OriginalPaper | Chapter

Umbilical-Type Surfaces in SpaceTime

Author : José M. M. Senovilla

Published in: Recent Trends in Lorentzian Geometry

Publisher: Springer New York

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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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previous chapter Calabi–Bernstein Results and Parabolicity of Maximal Surfaces in Lorentzian Product Spaces
next chapter Stability of Marginally Outer Trapped Surfaces and Applications
Footnotes
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
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
Notice the sign convention, which may not coincide with the preferred one for everybody.
 
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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Metadata
Title
Umbilical-Type Surfaces in SpaceTime
Author
José M. M. Senovilla
Copyright Year
2013
Publisher
Springer New York
Book
Recent Trends in Lorentzian Geometry

Print ISBN: 978-1-4614-4896-9

Electronic ISBN: 978-1-4614-4897-6

Copyright Year: 2013

DOI
https://doi.org/10.1007/978-1-4614-4897-6_3

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