Submanifolds of Euclidean space with parallel second fundamental form.

Vilms, Jaak

doi:10.1090/S0002-9939-1972-0290298-8

Submanifolds of Euclidean space with parallel second fundamental form.
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Proc. Amer. Math. Soc. 32 (1972), 263-267 Request permission

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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