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On the existence of spherically bent submanifolds, an analogue of a theorem of E. Cartan. (English) Zbl 1053.53014
In E. Cartan’s book Leçons sur la géométrie des espaces de Riemann, Gauthier-Villars, Paris (1951; Zbl 0044.18401) on can find a result providing a condition sufficient for a linear subspace \(V\subset T_pM\) to become the tangent space of a totally geodesic submanifold of \(M\). Here, the authors provide an analogous condition for so called spherical submanifolds: a submanifold \(N\) of \(M\) is said to be spherical whenever it is totally umbilical and its mean curvature vector is parallel in the normal bundle of \(N\). The condition in question is expressed in terms of the parallel transport and the curvature tensor of \(M\); it seems to be natural but complicated enough not to be provided here in details.

53B25 Local submanifolds
53C40 Global submanifolds
53B20 Local Riemannian geometry
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