Geodesic
The Global Structure of Locally Convex Hypersurfaces in Finsler--Hadamard Manifolds
Matematičeskie zametki, Tome 87 (2010) no. 2, pp. 163-174

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Locally convex compact immersed hypersurfaces in the Finsler–Hadamard space with bounded $T$-curvature are considered. Under certain conditions on normal curvatures, such hypersurfaces are proved to be convex, embedded, and homeomorphic to the sphere. To this end, the Rauch theorem is generalized to exponential maps of hypersurfaces and the convexity of parallel hypersurfaces is proved.
Keywords: Riemannian manifold, Rauch comparison theorem, Finsler metric, Gaussian, sectional, normal curvature, locally convex immersion, $T$-curvature, parallel hypersurface, Levi-Cività connection. 
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     author = {A. A. Borisenko and E. A. Olin},
     title = {The {Global} {Structure} of {Locally} {Convex} {Hypersurfaces} in {Finsler--Hadamard} {Manifolds}},
     journal = {Matemati\v{c}eskie zametki},
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A. A. Borisenko; E. A. Olin. The Global Structure of Locally Convex Hypersurfaces in Finsler--Hadamard Manifolds. Matematičeskie zametki, Tome 87 (2010) no. 2, pp. 163-174. http://geodesic.mathdoc.fr/item/MZM_2010_87_2_a0/