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- W2162296260 abstract "For an arbitrary n-dimensional Riemannian manifold N and an integer m ∈ {1,…,n−1} a covariant derivative $$hat{nabla}$$ on the Grassmann bundle ^ := G m (T N) is introduced which has the property that an m-dimensional submanifold M ⊂ N has parallel second fundamental form if and only if its Gauss map M → ^ is affine. (For N R n this result was already obtained by J. Vilms in 1972.) By means of this relation a generalization of Cartan's theorem on the total geodesy of a geodesic umbrella can be derived: Suppose, initial data (p,W,b) prescribing a tangent space W ∈ G m (T p N) and a second fundamental form b at p ∈ N are given; for these data we construct an m-dimensional ‘umbrella’ M = M(p,W,b) ⊂ N the rays of which are helical arcs of N; moreover, we present tensorial conditions (not involving $$hat{nabla}$$ ) which guarantee that the umbrella M has parallel second fundamental form. These conditions are as well necessary, and locally every submanifold with parallel second fundamental form can be obtained in this way. Mathematics Subject Classifications (2000): 53B25, 53B20, 53B21." @default.
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- W2162296260 date "2006-01-01" @default.
- W2162296260 modified "2023-09-25" @default.
- W2162296260 title "Submanifolds with Parallel Second Fundamental Form Studied via the Gauss Map" @default.
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- W2162296260 doi "https://doi.org/10.1007/s10455-006-1146-7" @default.
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