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 ≤ 1/(n-1) H² + 2c, n ≥ 4, or S ≤ 1/2 H² + min(2,(3p-3)/(2p-3))c, n = 3, then M is umbilical. This result generalizes the Okumura-Hasanis pinching theorem to the case of higher codimensions.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm98-2-5,
author = {Ziqi Sun},
title = {A pinching theorem on complete submanifolds with parallel mean curvature vectors},
journal = {Colloquium Mathematicae},
volume = {96},
year = {2003},
pages = {189-199},
zbl = {1113.53023},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm98-2-5}
}
Ziqi Sun. A pinching theorem on complete submanifolds with parallel mean curvature vectors. Colloquium Mathematicae, Tome 96 (2003) pp. 189-199. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm98-2-5/