Timeline for Are negatively pinched manifold locally conformally flat?
Current License: CC BY-SA 4.0
11 events
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| Mar 27, 2022 at 15:50 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
replaced the link to the arXiv front end; see https://meta.mathoverflow.net/questions/5124/is-it-time-to-replace-links-to-the-ucdavis-arxiv-frontend
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| Mar 24, 2022 at 0:43 | history | edited | David Roberts♦ | CC BY-SA 4.0 |
fixed arxiv front-end links; gave a title and attribution; replaced broken pdf link by Wayback Machine link
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| Sep 21, 2012 at 21:49 | comment | added | J. GE | Thanks Igor, I was thinking the difference of the Pontrjagin form is an exact form for conformal metrics but as you explained it is in fact the difference is zero, thanks! | |
| Sep 21, 2012 at 21:32 | comment | added | Igor Belegradek | @GB: "choose a flat connection?" There is no choice. We are on a Riemannian manifold where the Pontryagin class is represented by a Pontryagin form, which in turn is written in terms of components of the curvature tensor. It suffices to show the form vanishes pointwise. Fix a point, and consider its neighborhood which is conformally equivalent to an open subset of the Euclidean space. Since Pontryagin form is conformally invariant, and is zero for the Euclidean space, the form vanishes on the neighborhood. Which part is unclear? | |
| Sep 21, 2012 at 19:45 | comment | added | J. GE | @Igor, For locally conformally flat metric, one can choose a flat connection in a nbhd $U_x$ of each point $p\in X$ such that the Pontrjagin form vanishes at $U_x$, I don't see why the Pontryagin class vanishes globally. ps. I read Chern-Simons "Characteristic Forms and Geometric Invariants" Ann. 1974. On Theorem 4.5 they proved the conformal invariance for Pontrjagin class for globally conformal change of metric. Did I miss something? | |
| Sep 21, 2012 at 10:20 | comment | added | Igor Belegradek | @Misha: It is an interesting question whether there is a conformally flat version of Ontaneda's result. Somehow I am sceptical that Pontryagin classes is the only obstruction. | |
| Sep 21, 2012 at 10:17 | comment | added | Igor Belegradek | GB: here is the way I recall it. By Chern-Weil theory one can write Pontryagin forms (which represent the Pontryagin classes) in terms of components of the curvature tensor, and it turns out that if Weyl tensor vanishes, then so does the Pontryagin forms. I do not have a reference handy. This is a local computation, so as long as a manifold is locally conformally flat, the Pontryagin form vanishes. | |
| Sep 21, 2012 at 9:02 | comment | added | J. GE | @Igor, Is it true that "locally" conformally flat is enough for vanishing Pontryagin classes? I only know Pontryagin classes differs by exact forms for globally conformally metrics. | |
| Sep 21, 2012 at 4:41 | comment | added | Misha | On the other hand, some (maybe all?) of Gromov-Thurston examples admit conformally-flat metrics. Thus, it could be that Pontryagin classes are the only obstructions to existence of conformally-flat metrics on closed negatively curved manifolds which are sufficiently pinched. | |
| Sep 21, 2012 at 1:01 | history | edited | Igor Belegradek | CC BY-SA 3.0 |
added 58 characters in body
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| Sep 21, 2012 at 0:52 | history | answered | Igor Belegradek | CC BY-SA 3.0 |