A rigidity theorem for submanifolds inSn+p with constant scalar curvature
- 40 Downloads
LetM1 be a closed submanifold isometrically immersed in a unit sphereSn+p. Denote byR, H andS, the normalized scalar curvature, the mean curvature, and the square of the length of the second fundamental form ofM1, respectively. SupposeR is constant and ≥1. We study the pinching problem onS and prove a rigidity theorem forM1 immersed inSn+p with parallel normalized mean curvature vector field. Whenn≥8 or,n=7 andp≤2, the pinching constant is best.
Key wordsScalar curvature Mean curvature vector The second fundamental form
Unable to display preview. Download preview PDF.
- Chern, S.S., de Carmo, M., Kobayashi, S., 1970. Minimal Submanifolds of A Sphere with Second Fundamental Form of Constant Length. Functional A Analysis and Related Fields, p.59–75.Google Scholar
- Li, H.Z., 1994. Hypersurfaces with parallel mean curvature in a space forms.Math Ann,305:403–415.Google Scholar