Bulletin of the Australian Mathematical Society

Table of Contents - August 2004 - Volume 70, Issue 01  

Research Article

The curvature and topological properties of hypersurfaces with constant scalar curvature

Shu Shichanga1 and Liu Sanyanga2

a1 Department of Applied Mathematics, Xidian University, Xi'an 710071, Shaanxi, Peoples Republic of China, e-mail: xysxssc@yahoo.com.cn

a2 Department of Mathematics, Xianyang Teachers' University, Xianyang 712000, Shaanxi, Peoples Republic of China

In this paper, we consider n (n ≥ 3)-dimensional compact oriented connected hypersurfaces with constant scalar curvature n(n − 1)r in the unit sphere Sn+1(1). We prove that, if r ≥ (n − 2)/(n − 1) and S ≤ (n − 1)(n(r − 1) + 2)/(n − 2) + (n − 2)/(n(r − 1) + 2), then either M is diffeomorphic to a spherical space form if n = 3; or M is homeomorphic to a sphere if n ≥ 4; or M is isometric to the Riemannian product S0004972700035796_inline1, where c2 = (n − 2)/(nr) and S is the squared norm of the second fundamental form of M.

(Received September 30 2003)