Geodesic
On compact submanifolds of nonpositive external curvature in Riemannian spaces
Matematičeskie zametki, Tome 60 (1996) no. 1, pp. 3-10 
A. A. Borisenko. On compact submanifolds of nonpositive external curvature in Riemannian spaces. Matematičeskie zametki, Tome 60 (1996) no. 1, pp. 3-10. http://geodesic.mathdoc.fr/item/MZM_1996_60_1_a0/
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     author = {A. A. Borisenko},
     title = {On compact submanifolds of nonpositive external curvature in {Riemannian} spaces},
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     year = {1996},
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     url = {http://geodesic.mathdoc.fr/item/MZM_1996_60_1_a0/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper we consider compact multidimensional surfaces of nonpositive external curvature in a Riemannian space. If the curvature of the underlying space is $\ge1$ and the curvature of the surface is $\le1$, then in small codimension the surface is a totally geodesic submanifold that is locally isometric to the sphere. Under stricter restrictions on the curvature of the underlying space, the submanifold is globally isometric to the unit sphere.

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