Results in Mathematics

, Volume 63, Issue 1–2, pp 467–474 | Cite as

Biharmonic Submanifolds in a Riemannian Manifold with Non-Positive Curvature

Article

Abstract

In this paper, we show that, for every biharmonic submanifold (M, g) of a Riemannian manifold (N, h) with non-positive sectional curvature, if \({\int_M\vert \eta \vert^2 v_g < \infty}\) , then (M, g) is minimal in (N, h), i.e., \({\eta\equiv0}\), where η is the mean curvature tensor field of (M, g) in (N, h). This result gives an affirmative answer under the condition \({\int_M\vert \eta \vert^2 v_g < \infty}\) to the following generalized Chen’s conjecture: every biharmonic submanifold of a Riemannian manifold with non-positive sectional curvature must be minimal. The conjecture turned out false in case of an incomplete Riemannian manifold (M, g) by a counter example of Ou and Tang (in The generalized Chen’s conjecture on biharmonic sub-manifolds is false, a preprint, 2010).

Mathematics Subject Classification (2000)

58E20 

Keywords

Harmonic map Biharmonic map Isometric immersion Minimal Non-positive curvature 

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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Graduate School of Science and EngineeringYamaguchi UniversityYamaguchiJapan
  2. 2.Division of Mathematics, Graduate School of Information SciencesTohoku UniversitySendaiJapan
  3. 3.Institute for International EducationTohoku UniversitySendaiJapan

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