Timeline for non compact riemannian manifolds
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Nov 12, 2013 at 7:43 | comment | added | Anton Petrunin | Think first what do you mean by "classification". Say, the metrics with positive curvature on $\mathbb R^2$ are well understood, but for me this is not a "classification of these metrics". | |
| Aug 13, 2013 at 16:20 | comment | added | Thomas Richard | What is 'find all' ? It is possible to write a set of big and ugly formulas involving the coefficients $g_{ij}$ of the metric $g$ which will be equivalent to asking the curvature to be positive (and some other formulas saying that $g$ is complete), but I don't see why you could get anything better than that. Just look at rotationaly symmetric metrics $g=dr^2+f^2(r)d\theta^2$ on $\mathbb{R}^2$. The fact that $g$ is positively curved is equivalent to some inequality involving $f$ and its first and second derivatives. What more is there to say about those $f$ ? | |
| Aug 13, 2013 at 14:09 | comment | added | Jayq | @Thomas Richard: Actually, I want to know if it's possible to find all metrics of positive curvature on Euclidean space. | |
| Aug 13, 2013 at 13:09 | comment | added | Thomas Richard | What do you mean by classifying ? Being positively curved is an open condition for metrics (with the $C^2$ topology), and the result you quote shows that non compact positively curved manifolds are classified up to diffeomorphism. Can one expet something more ? | |
| Aug 13, 2013 at 8:26 | comment | added | Willie Wong | Cross posted to MSE: math.stackexchange.com/questions/461691/… | |
| Aug 13, 2013 at 7:35 | history | edited | Jayq | CC BY-SA 3.0 |
added 20 characters in body
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| Aug 7, 2013 at 17:01 | comment | added | Ben McKay | A paraboloid is a nice example: $y=x_1^2+x_2^2+\dots+x_n^2$. Intuitively, that should be something like what they all look like, as their Ricci curvature must decay to zero when you go far away from any chosen point. | |
| Aug 7, 2013 at 16:59 | comment | added | Ben McKay | You mean to classify non-compact Riemannian manifolds of positive sectional curvature, not just to classify non-compact Riemannian manifolds | |
| Aug 7, 2013 at 13:52 | history | edited | Peter Michor | CC BY-SA 3.0 |
Improved TeX and formatting
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| Aug 7, 2013 at 13:51 | review | First posts | |||
| Aug 7, 2013 at 13:55 | |||||
| Aug 7, 2013 at 13:35 | history | asked | Jayq | CC BY-SA 3.0 |