Abstract
Let f:M→ℝ be an isometric immersion of an m-dimensional Riemannian manifold M into the n-dimensional Euclidean space. Its Gauss map g:M→G m (ℝn) into the Grassmannian G m (ℝn) is defined by assigning to every point of M its tangent space, considered as a vector subspace of ℝn. The third fundamental form b of f is the pull-back of the canonical Riemannian metric on G m (ℝn) via g. In this article we derive a complete classification of all those f (with flat normal bundle) for which the Gauss map g is homothetical; i.e. b is a constant multiple of the Riemannian metric on M. Using these results we furthermore classify all those f (with flat normal bundle) for which the third fundamental form b is parallel w.r.t. the Levi-Civita connection on M.
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Nölker, S. Isometric immersions with homothetical Gauss map. Geom Dedicata 34, 271–280 (1990). https://doi.org/10.1007/BF00181689
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DOI: https://doi.org/10.1007/BF00181689