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Foliations of doubly warped products by k-umbilical hypersurfaces

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Abstract

A k-umbilical hypersurface of a \((n+1)\)-dimensional semi-Riemannian manifold has k equal principal curvatures at each point. We study conditions under which a Lorentzian manifold admits a foliation by such hypersurfaces. Originally motivated by the existence of particular foliations of the Schwarzschild black hole, we extend our theory to the class of manifolds given as doubly warped products. The key ingredient for our study is a new class of vector fields which includes conformal vector fields. After obtaining general results about manifolds admitting such kind of vector fields, we give a characterization of the Schwarzschild black hole in the class of Lorentzian manifolds admitting a pair of such vector fields and satisfying some boundary conditions .

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Funding

A. G. Colares was partially supported by the National Council of Scientific and Technological Development (CNPq), Brasília, Brazil, grant number 311535/2017-8. O. Palmas was partially supported by DGAPA-UNAM, under projects IN113516 and IN115119.

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  1. A. Gervasio Colares contributed equally to this work.

Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal do Ceará, Av. Humberto Monte s/n, Campus do Pici, Bloco 914, Fortaleza, Ceará, 60.455-760, Brazil

    A. Gervasio Colares

  2. Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, Circuito Exterior, Ciudad Universitaria, 04510, Mexico City, Ciudad de México, México

    Oscar Palmas

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  1. A. Gervasio Colares
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Colares, A.G., Palmas, O. Foliations of doubly warped products by k-umbilical hypersurfaces. Annali di Matematica 201, 2261–2285 (2022). https://doi.org/10.1007/s10231-022-01199-6

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  • DOI: https://doi.org/10.1007/s10231-022-01199-6

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