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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
  • Sadahiro Maeda
  • 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 space forms, parallel isometric immersions, isotropic immersions, totally umbilic, Veronese manifolds, sectional curvatures, parallel mean curvature vector
MSC Classifications: 53C40, 53C42 show english descriptions Global submanifolds [See also 53B25]
Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]
53C40 - Global submanifolds [See also 53B25]
53C42 - Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]
 

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