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I am trying to prove the hyperbolic space is complete. It looks one need to apply Hopf-Rinow theorem, but I don't know what to start. More precisely, I don't know what is a good way to show the exponential map is defined on the entire $T_{p}M$.

Another approach probably be proving that with the hyperboloid model $x_{0}^{2}-x_{1}^{2} \cdots -x_{n}^{2}=1$ is a complete metric space under the distance $d(\textbf{x},\textbf{y})=arcosh (x_{0}y_{0}-x_{1}y_{1}-\cdots-x_{n}y_{n})$.

Any help is appreciated!

finiteness
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  • Do you mean topologicaly complete or geodesicaly complete. I don't really now hyperbolic geometry. But on the wikipedia page, it is said that the hyperbolic space $H^n$ is diffeomorphic to $\mathbb{R}^n$. So it is topologicaly complete. –  Oct 03 '18 at 00:17
  • I mean geodesically complete. – finiteness Oct 03 '18 at 18:20
  • It seems that Hopf-Rinow theorem says that topologicaly complete and geodesically complete are equivalent. –  Oct 03 '18 at 18:43
  • Why completeness is preserved under diffeomorphism? – finiteness Oct 03 '18 at 19:30
  • You're right, I'm just silly ! –  Oct 03 '18 at 20:13

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