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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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Given any nonempty, complete geodesic submanifold $M$ of $n$-dimensional hyperbolic space. Pick a point $x\in M$ and consider the subspace $T_xM\subset T_\mathbb{H}^n$. Now you could show that $M=exp(T_xM)$ since $M$ is complete. Thus $M$ is uniquely determined by that vectorspace. Given any subspace $V\subset T_x\mathbb{H}^n$ at some point $x$ try to find (among the submanifolds that you mentioned) one with $T_xM=V$. This shows that these are really all geodesic (complete) submanifolds. – HenrikRueping Nov 27 '12 at 10:56

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