I have a question to find all geodesic submanifolds of the hyperbolic space in n-dim. I did an exercise that all geodesics must be either lines perpendicular to the boundary of the hyperbolic space (under half space model) or great circles of spheres centered at the boundary. But I am not sure how to prove that only planes perpendicular to the boundary and spheres perpendicular to the boundary the only manifolds. Could you please help?
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