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

Published online by Cambridge University Press:  20 November 2018

Sadahiro Maeda  and
Seiichi Udagawa
Sadahiro Maeda
Affiliation:
(Maeda) Department of Mathematics, Shimane University, Matsue 690-8504, Japan Current address: Department of Mathematics, Saga University, 1 Honzyo, Saga 840-8502, Japan, smaeda@ms.saga-u.ac.jp
Seiichi Udagawa
Affiliation:
(Udagawa) Department of Mathematics, School of Medicine, Nihon University, Itabashi, Tokyo 173-0032, Japan, sudagawa@med.nihon-u.ac.jp
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Abstract

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For an isotropic submanifold ${{M}^{n}}(n\underline{\underline{>}}3)$ of a space form ${{\tilde{M}}^{n+p}}(c)$ 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 ${{\tilde{M}}^{n+p}}$ is parallel and our manifold ${{M}^{n}}$ is a space form.

Keywords

53C4053C42space formsparallel isometric immersionsisotropic immersionstotally umbilicVeronese manifoldssectional curvaturesparallel mean curvature vector
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 61 , Issue 3 , 01 June 2009 , pp. 641 - 655
Copyright
Copyright © Canadian Mathematical Society 2009

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