Abstract
Let M, resp.\(\tilde M\), denote Riemannian manifolds of dimensions m>4, resp.\(\tilde m\)=m+2, and of constant sectional curvatures C, resp.\(\tilde C\), with\(\tilde C\)<C, and suppose that M is complete (i.e. M is a space form). Let f:M→\(\tilde M\) denote an isometric immersion. Then the open subset of M consisting of all non-umbilic points of f is foliated by complete hypersurfaces of M, which are umbilical both in M and in\(\tilde M\). In this paper we study this foliation L in detail. In particular we prove: If in addition C>0 and\(\tilde M\) is a standard space form, then the foliation L is a (globally) trivial fibre bundle with fibre Sm−1.
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Henke, W. Isometrische Immersionen der Kodimension 2 von Raumformen. Manuscripta Math 19, 165–188 (1976). https://doi.org/10.1007/BF01275420
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DOI: https://doi.org/10.1007/BF01275420