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Submanifolds of Euclidean space with parallel second fundamental form.


Author: Jaak Vilms
Journal: Proc. Amer. Math. Soc. 32 (1972), 263-267
MSC: Primary 53.74
DOI: https://doi.org/10.1090/S0002-9939-1972-0290298-8
MathSciNet review: 0290298
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Abstract: In this paper necessary conditions are given for a complete Riemannian manifold $ M^{n}$ to admit an isometric immersion into $ R^{n+p}$ with parallel second fundamental form. Namely, it is shown that $ M^{n}$ must be affinely equivalent either to a totally geodesic submanifold of the Grassmann manifold $ G(n,p)$, or to a fibre bundle over such a submanifold, with Euclidean space as fibre and the structure being close to a product. (An affine equivalence is a diffeomorphism that preserves Riemannian connections.) The proof depends on the auxiliary result that the second fundamental form is parallel iff the Gauss map is a totally geodesic map.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0290298-8
Keywords: Submanifold of Euclidean space, second fundamental form, totally geodesic submanifold, totally geodesic map, Grassmann manifold, Gauss map, relative nullity
Article copyright: © Copyright 1972 American Mathematical Society

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