The Global Structure of Locally Convex Hypersurfaces in Finsler–Hadamard Manifolds

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.