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A pinching theorem on complete submanifolds with parallel mean curvature vectors

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Sun Z.

Colloquium Mathematicum

2003

tom 98

nr 2

189-199

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.

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1 Colloquium Mathematicum

1 Sun Z.

1 2003

Wyniki wyszukiwania

A pinching theorem on complete submanifolds with parallel mean curvature vectors