Timeline for Can the exponential map be used to define geodesics (and hence, generalisations of geodesics)?
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 27, 2023 at 23:08 | vote | accept | AmorFati | ||
| Dec 4, 2022 at 23:30 | answer | added | user44143 | timeline score: 1 | |
| Dec 4, 2022 at 16:00 | answer | added | Vladimir S Matveev | timeline score: 7 | |
| Dec 2, 2022 at 23:57 | comment | added | Thomas Rot | I think spray to search for in this context | |
| Dec 2, 2022 at 22:51 | comment | added | Deane Yang | I don't know if this would be sufficient, but you could try something like the following, assuming $M$ is complete: 1) If $\gamma(t) = \exp_p(tv)$, then $\gamma'(0) = v$ 2) If $q = exp_p(v_p)$ and $v_q$ is the parallel transpoart of $v_p$ to $T_qM$ along the curve $t\mapsto\exp_p(tv)$, then $\exp_q(-v_q) = p$. 3) Given $p \in M$ there exists a neighborhood $B$ of $0 \in T_pM$ such that if $v \in B$, then $d(p,\exp_p(v)) = |v|$. | |
| Dec 2, 2022 at 21:40 | comment | added | Carlos Esparza | One way to generalize geodesics is replacing the Levi-Civita connection by any other (torsion-free if you want) connection on $TM$. There will also be a corresponding notion of exponential map. I suspect you might be able to extract a connection out of your family of maps $\sigma_p$ by pulling vector fields back along $\sigma_p$, differentiating at $0 \in T_ M$ and pushing forward along $\sigma_p$. | |
| Dec 2, 2022 at 21:24 | history | asked | AmorFati | CC BY-SA 4.0 |