%0 Journal Article
%T Hypersurfaces with Two Distinct Para-Blaschke Eigenvalues in
%A Junfeng Chen
%A Shichang Shu
%J International Journal of Mathematics and Mathematical Sciences
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/398746
%X Let be an -dimensional immersed hypersurface without umbilical points and with vanishing M£¿bius form in a unit sphere , and let and be the Blaschke tensor and the M£¿bius second fundamental form of , respectively. We define a symmetric tensor which is called the para-Blaschke tensor of , where is a constant. An eigenvalue of the para-Blaschke tensor is called a para-Blaschke eigenvalue of . The aim of this paper is to classify the oriented hypersurfaces in with two distinct para-Blaschke eigenvalues under some rigidity conditions. 1. Introduction In the M£¿bius geometry of hypersurfaces, Wang [1] studied invariants of hypersurfaces in a unit sphere under the M£¿bius transformation group. Let be an -dimensional immersed hypersurface without umbilical points in . We choose a local orthonormal basis for the induced metric with dual basis . Let be the second fundamental form of the immersion and the mean curvature of the immersion . By putting , the M£¿bius metric of the immersion is defined by which is a M£¿bius invariant. , , and are called the M£¿bius form, the Blaschke tensor, and the M£¿bius second fundamental formof the immersion , respectively (see [1]), where and and are the Hessian matrix and the gradient with respect to the induced metric . It was proved by [1] that , , and are M£¿bius invariants. In the study of the M£¿bius geometry of hypersurfaces, one of the important aims is to characterize hypersurfaces in terms of M£¿bius invariants. Concerning this topic, there are many important results; one can see [2¨C9]. We should notice that [5] classified all umbilic-free hypersurfaces with parallel M£¿bius second fundamental form. Recently, by making use of the two important M£¿bius invariants, the Blaschke tensor and the M£¿bius second fundamental form of , Zhong and Sun [10] defined a symmetric tensor which is called the para-Blaschke tensor of , where is a constant. An eigenvalue of the para-Blaschke tensor is called a para-Blaschke eigenvalue of . In [7], Li and Wang investigated and completely classified hypersurfaces without umbilical points and with vanishing M£¿bius form in , which satisfy . It should be noted that the condition implies that the para-Blaschke eigenvalues of are all equal. If has two distinct constant para-Blaschke eigenvalues, Zhong and Sun [10] obtained the following. Theorem 1 (see [10]). Let be an -dimensional immersed hypersurface without umbilical points. If is of two distinct constant para-Blaschke eigenvalues and of vanishing M£¿bius form , then is locally M£¿bius equivalent to(1)the Riemannian product in , or(2)the image of of the
%U http://www.hindawi.com/journals/ijmms/2014/398746/