Timeline for Closed geodesics on a closed, negatively curved Riemannian manifold
Current License: CC BY-SA 3.0
7 events
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| Dec 18, 2012 at 3:00 | comment | added | Clark | Thank you, I had bad luck finding this with a few Riemannian geometry texts and I flipped through do Carmo's book and somehow missed this result, which is proven in the text. | |
| Dec 18, 2012 at 1:22 | history | edited | Rami |
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| Dec 17, 2012 at 21:51 | comment | added | Ian Agol | This question is probably more appropriate for Math.stackexchange - you can find the answer in many Riemannian geometry textbooks. | |
| Dec 17, 2012 at 19:01 | history | edited | Pablo Lessa |
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| Dec 17, 2012 at 8:04 | comment | added | Misha | This is an easy corollary of the "flat strip" theorem: If $X$ is a CAT(0) space and $g_1, g_2$ are two geodesics in $X$ which are Hausdorff-close, then $g_1, g_2$ cobound a flat strip in $X$. You can also take a look at do Carmo's book "Riemannian geometry", I think there is a proof there. | |
| Dec 17, 2012 at 8:04 | comment | added | Dan Fox | This result is a special case of the following more general result of Eells and Sampson (see Theorem B in the last section of "Harmonic mappings of Riemannian manifolds") that any continuous map between closed Riemannian manifolds $M$ and $N$ where $M$ has nonnegative Ricci curvature and $N$ has nonpositive sectional curvature is homotopic to a totally geodesic map; if, moreover, $N$ has somewhere strictly negative curvature, then the map is either null-homotopic or homotopic to a closed geodesic. Any survey on harmonic mappings should contain some version of this. | |
| Dec 17, 2012 at 7:05 | history | asked | Clark | CC BY-SA 3.0 |