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March 2009 On submanifolds with parallel mean curvature vector
Kellcio O. Araújo, Keti Tenenblat
Kodai Math. J. 32(1): 59-76 (March 2009). DOI: 10.2996/kmj/1238594546

Abstract

We consider Mn, n ≥ 3, a complete, connected submanifold of a space form $\tilde{M}^{n+p}(\tilde{c})$, whose non vanishing mean curvature vector H is parallel in the normal bundle. Assuming the second fundamental form h of M satisfies the inequality <h>2n2 |H|2/(n - 1), we show that for $\tilde{c}$ ≥ 0 the codimension reduces to 1. When M is a submanifold of the unit sphere, then Mn is totally umbilic. For the case $\tilde{c}$ < 0, one imposes an additional condition that is trivially satisfied when $\tilde{c}$ ≥ 0. When M is compact and has non-negative Ricci curvature then it is a geodesic hypersphere in the hyperbolic space. An alternative additional condition, when $\tilde{c}$ < 0, reduces the codimension to 3.

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Kellcio O. Araújo. Keti Tenenblat. "On submanifolds with parallel mean curvature vector." Kodai Math. J. 32 (1) 59 - 76, March 2009. https://doi.org/10.2996/kmj/1238594546

Information

Published: March 2009
First available in Project Euclid: 1 April 2009

zbMATH: 1160.53026
MathSciNet: MR2518554
Digital Object Identifier: 10.2996/kmj/1238594546

Rights: Copyright © 2009 Tokyo Institute of Technology, Department of Mathematics

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Vol.32 • No. 1 • March 2009
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