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$r$-mean curvatures

Proceedings of the Edinburgh Mathematical Society (1)

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INTEGRAL FORMULAE FOR SPACELIKE HYPERSURFACES IN CONFORMALLY STATIONARY SPACETIMES AND APPLICATIONS
Luis J. Alías, Aldir Brasil, A. Gervasio Colares
Journal:

Proceedings of the Edinburgh Mathematical Society / Volume 46 / Issue 2 / June 2003

Published online by Cambridge University Press:

04 July 2003, pp. 465-488
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In this paper we develop general Minkowski-type formulae for compact spacelike hypersurfaces immersed into conformally stationary spacetimes, that is, Lorentzian manifolds admitting a timelike conformal field. We apply them to the study of the umbilicity of compact spacelike hypersurfaces in terms of their $r$-mean curvatures. We derive several uniqueness results, for instance, compact spacelike hypersurfaces are umbilical if either some of their $r$-mean curvatures are linearly related or one of them is constant.
AMS 2000 Mathematics subject classification: Primary 53C42. Secondary 53B30; 53C50; 53Z05; 83C99