Geodesic
On compact submanifolds of nonpositive external curvature in Riemannian spaces
Matematičeskie zametki, Tome 60 (1996) no. 1, pp. 3-10.

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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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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/

[1] Borisenko A. A., “Ob ekstremalnykh svoistvakh kompaktnykh parabolicheskikh poverkhnostei v rimanovom prostranstve”, Matem. sb., 133 (175):1 (1987), 112–126

[2] Borisenko A. A., “O vneshne geometricheskikh svoistvakh parabolicheskikh poverkhnostei i topologicheskikh svoistvakh sedlovykh poverkhnostei v simmetricheskikh prostranstvakh ranga 1”, Matem. sb., 116 (158):3 (1981), 440–457 | MR | Zbl

[3] Eizenkhart L., Rimanova geometriya, IL, M., 1948

[4] Ferus D., “Totally geodesic foliations”, Math. Ann., 88:4 (1970), 313–317 | DOI | MR

[5] Frankel T., “Manifolds with positive curvature”, Pacif. J. Math., 11 (1961), 165–174 | MR | Zbl

[6] Toponogov V. A., “Nekotorye ekstremalnye teoremy rimanovoi geometrii”, Sib. matem. zh., 10:2 (1967), 459–467