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On the linear Weingarten spacelike submanifolds immersed in a locally symmetric semi-Riemannian space

  • Jogli G. Araújo
  • Weiller F. C. Barboza
  • Henrique F. de LimaEmail author
  • Marco Antonio L. Velásquez
Original Paper
  • 13 Downloads

Abstract

Let \(M^{n}\) be an n-dimensional complete linear Weingarten spacelike submanifold immersed with parallel normalized mean curvature vector field and flat normal bundle in a locally symmetric semi-Riemannian space \(L_{p}^{n+p}\) of index p, which obeys standard curvature constraints (such an ambient space can be regarded as an extension of a semi-Riemannian space form). In this setting, our purpose is to establish sufficient conditions guaranteeing that such a spacelike submanifold \(M^{n}\) be either totally umbilical or isometric to an isoparametric hypersurface of a totally geodesic submanifold \(L_{1}^{n+1}\hookrightarrow L_{p}^{n+p}\), with two distinct principal curvatures, one of which is simple. Our approach is based on a suitable Simons type formula jointly with a version of the Omori–Yau’s generalized maximum principle for a Cheng–Yau’s modified operator.

Keywords

Locally symmetric semi-Riemannian space Complete linear Weingarten spacelike submanifolds Parallel normalized mean curvature vector field Conformally flat manifolds 

Mathematics Subject Classification

Primary 53C42 Secondary 53A10 53C20 53C50 

Notes

Acknowledgements

The authors would like to thank the referee for his/her valuable suggestions and useful comments which improved the paper. The second author is partially supported by CAPES, Brazil. The third and fourth authors are partially supported by CNPq, Brazil, Grants 303977/2015-9 and 311224/2018-0, respectively.

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Copyright information

© The Managing Editors 2019

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade Federal Rural de PernambucoRecifeBrazil
  2. 2.Departamento de MatemáticaUniversidade Federal de Campina GrandeCampina GrandeBrazil

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