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Isometric immersions with homothetical Gauss map

✍ Scribed by Stefan Nölker

Publisher
Springer
Year
1990
Tongue
English
Weight
431 KB
Volume
34
Category
Article
ISSN
0046-5755

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✦ Synopsis

Let f: M ~ R be an isometric immersion of an m-dimensional Riemannian manifold M into the n-dimensional Euclidean space. Its Gauss map 0: M ~ G=(R") into the Grassmannian G,.(R") is defined by assigning to every point of M its tangent space, considered as a vector subspace of R". The third fundamental form b off is the pull-back of the canonical Riemannian metric on GIn(R") via 0. In this article we derive a complete classification of all thosef (with flat normal bundle) for which the Gauss map e is homothetical; i.e. b is a constant multiple of the Riemannian metric on M. Using these results we furthermore classify all thosef (with fiat normal bundle) for which the third fundamental form b is parallel w.r.t, the Levi-Civita connection on M.

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