Timeline for Maps on Riemannian manifold agreeing with geodesics
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Sep 6, 2021 at 17:38 | comment | added | Robert Bryant | Actually, I believe that the original proof that there is a local coordinate system in which the geodesics are straight lines if and only if the Gauss curvature is constant is due to Beltrami. The proof in higher dimensions follows immediately from the proof in dimension $2$. For a straightforward elementary proof in dimension $2$, you can see Exercise 13 in Chapter 4, Section 6 of do Carmo's Differential Geometry of Curves and Surfaces. | |
| Sep 6, 2021 at 17:02 | comment | added | Kapil | One (somewhat convoluted) proof of the result for dimension $\geq 3$ can be found here on pages 12-13. The proof for dimension 2 is perhaps in the book by Sharpe; as Ben McKay has said it is about projectively flat connections. | |
| Sep 6, 2021 at 13:23 | comment | added | John Rached | I believe OP is trying to get clarification on where constant curvature of the metric comes in | |
| Sep 6, 2021 at 11:13 | comment | added | Ben McKay | Your condition is that the projective connection induced from the Levi-Civita connection is flat. You can probably find a proof of this in Projective Differential Geometry Old and New by Tabachnikov and Ovsienko. | |
| Sep 6, 2021 at 7:55 | comment | added | Kacper Kurowski | Yes, though, if possible, could you provide me with a reference to the result you mention? I tried to find it and I mostly find the Cartan-Hadamard theorem regarding the complete Riemannian manifolds with non-positive sectional curvature. Also, if that changes anything, I only want to map line segments from some neighbourhood to geodesics defined on $[0,1]$. | |
| Sep 6, 2021 at 2:14 | comment | added | Kapil | Don't you want $\phi$ to be a a diffeomorphism? If so, then you are asking whether there is a diffeomorphism that takes geodesics to lines (and vice versa). This requires $M$ to have constant curvature. (Essentially, this is a result of Cartan and Hadamard in dimension $\geq 3$ and of Blaschke in dimension 2.) | |
| Sep 5, 2021 at 22:52 | history | asked | Kacper Kurowski | CC BY-SA 4.0 |