On the bi-Ricci curvature and some applications

• Autores: Li Yao Wen, Zou Xiao Rong
• Localización: Houston journal of mathematics, ISSN 0362-1588, Vol. 34, Nº 2, 2008, págs. 467-482
• Idioma: inglés
• Texto completo no disponible (Saber más ...)
• Resumen

  • In this paper, we investigate the properties of the bi-Ricci curvature of a Riemannian manifold and use this geometrical quantity to study submanifolds in two ways. First, we shall prove a sharp lower bound of the bi-Ricci curvature of an immersed submanifold in a general Riemannian manifold and use the estimation to characterize the Clifford hypersurface S2((1-c²)1/2)× Sn-2(c) in the standard sphere Sn+1(1). Secondly, we shall prove that there are no nontrivial L2 harmonic 1-forms on a strongly stable hypersurface M of a general Riemannian manifold N when the bi-Ricci curvature of N is no less than certain lower bound, which gives a topological obstruction for the stability of M. The result about the nonexistence of nontrivial L2 harmonic forms on the strongly stable hypersurface M is also used to study the number of its ends.