Timeline for Exponential map and covariant derivative
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
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| May 11, 2010 at 4:12 | comment | added | Ben Whale | Lucas is right, $\exp$ is a diffeomorphism from a subset of the tangent space onto it's image. Yes in general $\exp_p(U)\neq M$. This is the case even when $U=T_pM$. To see this take a look at the Hopf-Rinow Theorem and consider the case of $S^2$, which we know can't be covered by a single chart. For interests sake there is no corresponding Hopf-Rinow theorem for Lorentzian manifolds and in general the three equivalent conditions of the Hopf-Rinow theorem are independent for Lorentzian metrics. | |
| May 10, 2010 at 21:52 | comment | added | user6011 | So ${exp}_{P}(U) \ne M$. | |
| May 10, 2010 at 21:36 | comment | added | Lucas Kaufmann | It is a diffeomorphism from a neighbourhood of $0 \in T_p M$ onto its image. You can see this by noting that the derivative of $exp_p$ at 0 is the indentity map and using the inverse function theorem. | |
| May 10, 2010 at 19:51 | comment | added | user6011 | Ben, thank you very much! In your answer, you used `` homeomorphism" several times. I would like to know that '${exp}_{p}: U\in {T}_{p}M \rightarrow M$´ is really a homeomorphism? | |
| May 10, 2010 at 13:34 | history | answered | Ben Whale | CC BY-SA 2.5 |