Abstract: Let Mⁿ be a closed submanifold isometrically immersed in a unit sphere S^n+p. Denote by R, H and S, the normalized scalar curvature, the mean curvature, and the square of the length of the second fundamental form of Mⁿ, respectively. Suppose R is constant and ≥1. We study the pinching problem on S and prove a rigidity theorem for Mⁿ immersed in S^n+p with parallel normalized mean curvature vector field. When n≥8 or, n=7 and p≤2, the pinching constant is best.

Key words: Scalar curvature, Mean curvature vector, The second fundamental form

Please provide your name, email address and a comment

Your Name: Affili: E-Mail Address:

Comment (Please comment in English):


Verification Code(click to change a pic):