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# Characterization of Parallel Isometric Immersions of Space Forms into Space Forms in the Class of Isotropic Immersions

Published:2009-06-01
Printed: Jun 2009
• Seiichi Udagawa
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## Abstract

For an isotropic submanifold $M^n\,(n\geqq3)$ of a space form $\widetilde{M}^{n+p}(c)$ of constant sectional curvature $c$, we show that if the mean curvature vector of $M^n$ is parallel and the sectional curvature $K$ of $M^n$ satisfies some inequality, then the second fundamental form of $M^n$ in $\widetilde{M}^{n+p}$ is parallel and our manifold $M^n$ is a space form.
 Keywords: space forms, parallel isometric immersions, isotropic immersions, totally umbilic, Veronese manifolds, sectional curvatures, parallel mean curvature vector
 MSC Classifications: 53C40 - Global submanifolds [See also 53B25] 53C42 - Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]

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