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Characterizations of complete hypersurfaces in locally symmetric Riemannian manifolds

Volume 153 / 2018

Jogli G. Araújo, Márcio Batista, Henrique F. de Lima Colloquium Mathematicum 153 (2018), 149-162 MSC: Primary 53C42; Secondary 53A10, 53C20, 53C50. DOI: 10.4064/cm7047-5-2017 Published online: 16 April 2018

Abstract

We deal with complete linear Weingarten hypersurfaces immersed in a locally symmetric Riemannian manifold whose sectional curvature obeys certain standard constraints. Under an appropriate restriction on the norm of the traceless part of the second fundamental form, we show that such a hypersurface must be either totally umbilical or an isoparametric hypersurface with two distinct principal curvatures, one of which is simple. Our approach is based on the use of a suitable Simons type formula combined with applications of some generalized maximum principles.

Authors

  • Jogli G. AraújoDepartamento de Matemática
    Universidade Federal de Campina Grande
    58.429-970 Campina Grande, Paraíba, Brazil
    e-mail
  • Márcio BatistaInstituto de Matemática
    Universidade Federal de Alagoas
    57.072-970 Maceió, Alagoas, Brazil
    e-mail
  • Henrique F. de LimaDepartamento de Matemática
    Universidade Federal de Campina Grande
    58.429-970 Campina Grande, Paraíba, Brazil
    e-mail

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