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Sur certaines champs de sous-espaces vectoriels et sur les sous- variétés riemanniennes dont la connexion normale est plate. (French) Zbl 0534.53045
Let V be a p-dimensional submanifold of a Riemannian manifold \(\tilde V\). In a tubular neighbourhood of V one defines various p-dimensional distributions (or vector subspace fields). One is the distribution of subspaces orthogonal to the exponentiated normal disks of V in \(\tilde V\). Another is obtained by parallel translating the tangent spaces of V along normal geodesics. A third is defined in terms of the values of Jacobi fields, and a fourth by means of the second fundamental form. Manifolds V of constant curvature are characterized by properties of these distributions, and the integrability of certain of these distributions is closely related to flatness of the normal connection of V. Finally, an estimate for the absolute value of the Gauss-Bonnet curvature of an even-dimensional V with flat normal connection is obtained when \(\tilde V\) has constant sectional curvature.
Reviewer: J.Hebda

MSC:
53C40 Global submanifolds
53C20 Global Riemannian geometry, including pinching
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