A pinching theorem on complete submanifolds with parallel mean curvature vectors

Ziqi Sun¹

¹ Department of Mathematics and Statistics Wichita State University Wichita, KS 67226, U.S.A.

Colloquium Mathematicum, Tome 98 (2003) no. 2, pp. 189-199

Résumé

Let $M$ be an $n$-dimensional complete immersed submanifold with parallel mean curvature vectors in an $(n+p)$-dimensional Riemannian manifold $N$ of constant curvature $c>0$. Denote the square of length and the length of the trace of the second fundamental tensor of $M$ by $S$ and $H$, respectively. We prove that if $$ S\leq\frac{1}{n-1}\,H^2+2c,\ \quad n\geq 4, $$ or $$ S\leq\frac{1}{2} \, H^2 + \min\bigg(2,\frac{3p-3}{2p-3}\bigg)c,\ \quad n=3, $$ then $M$ is umbilical. This result generalizes the Okumura–Hasanis pinching theorem to the case of higher codimensions.

Détail

DOI : 10.4064/cm98-2-5

Keywords: n dimensional complete immersed submanifold parallel mean curvature vectors dimensional riemannian manifold constant curvature denote square length length trace second fundamental tensor respectively prove leq frac n quad geq leq frac min bigg frac p p bigg quad umbilical result generalizes okumura hasanis pinching theorem higher codimensions