Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
2  
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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.