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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
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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  • [1] J. Cheeger and D. Gromoll, The structure of complete manifolds of nonnegative curvature, Bull. Amer. Math. Soc. 74 (1968), 1147-1150. MR 38 #635. MR 0232310 (38:635)
  • [2] S.-S. Chern and N. Kuiper, Some theorems on the isometric imbedding of compact Riemann manifolds in Euclidean space, Ann. of Math. (2) 56 (1952), 422-430. MR 14, 408. MR 0050962 (14:408e)
  • [3] J. Eells, Jr. and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109-160. MR 29 #1603. MR 0164306 (29:1603)
  • [4] S. Kobayashi and K. Nomizu, Foundations of differential geometry, Vol. 2, Interscience Tracts in Pure and Appl. Math., no. 15, vol. 2, Interscience, New York, 1969. MR 38 #6501
  • [5] O. Loos, Symmetric spaces. I : General theory, Benjamin, New York, 1969. MR 39 #365a. MR 0239005 (39:365a)
  • [6] E. Ruh and J. Vilms, The tension field of the Gauss map, Trans. Amer. Math. Soc. 149 (1970), 569-573. MR 41 #4400. MR 0259768 (41:4400)
  • [7] J. Vilms, Totally geodesic maps, J. Differential Geometry 4 (1970), 73-79. MR 41 #7589. MR 0262984 (41:7589)

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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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