Hypersurfaces with Two Distinct Para-Blaschke Eigenvalues in

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 [2–9]. 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