Timeline for Positive sectional curvature does not imply positive definite curvature operator?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Aug 17, 2022 at 3:20 | comment | added | C.F.G | mathoverflow.net/a/390636/90655 | |
| Aug 16, 2022 at 21:17 | comment | added | Renato G. Bettiol | @C.F.G: Some definitions might be more obvious to some of us :) In my opinion, this is an interesting and useful definition because it is algebraically very natural (some reasons are given in Thm B in arxiv.org/pdf/1908.03713.pdf cf. the difference between nonnegative polynomials and polynomials which are sums of squares), sits strictly between a class of manifolds still poorly understood (sec>0) and one completely classified (pos-def curvature operator), and was crucially used to produce the latest example in the former class (arxiv.org/abs/0809.2304). [Sorry for late reply!] | |
| Aug 16, 2022 at 21:10 | comment | added | Renato G. Bettiol | @CarloMantegazza: Any manifold with sec>0 whose universal cover is not isometric to a product of (i) Euclidean space, (ii) sphere with metric of positive-semidef curvature operator, (iii) compact irreducible symm space, (iv) compact Kaehler mfld biholomorphic to $CP^n$ with positive-semidef curature operator on (1,1)-forms. For example, the Wallach flag manifolds $W^6$, $W^{12}$, $W^{24}$, which are even homogeneous (see e.g. Ziller's survey Sec 4: www2.math.upenn.edu/~wziller/papers/…) [Sorry for the super late answer!!] | |
| Apr 29, 2022 at 11:28 | comment | added | C.F.G | @RenatoG.Bettiol: Isn't "strongly nonnegative curvature" or "strongly positive curvature" an obvious definition that at least from top view is a useless tool? To me it is like this that I have a negative number in mind and you are seeking for a positive number that has a positive or nonnegative result when add up to my number. Why should this be a useful tool? | |
| Apr 1, 2021 at 2:08 | comment | added | Carlo Mantegazza | @Renato: what is an example of a manifold with positive sec and curvature operator with some negative eigenvalues? | |
| Mar 15, 2021 at 9:37 | history | edited | C.F.G | CC BY-SA 4.0 |
format improved using MO citation tool
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| Mar 18, 2017 at 17:47 | comment | added | Renato G. Bettiol | @Igor: not that I am aware of... Also Werner Ballmann asked me a few months ago whether strongly positive curvature could be seen as an intermediate pinching. But, just like the case for the curvature operator, there doesn't seem to be too much one can say. | |
| Mar 18, 2017 at 0:53 | comment | added | Igor Belegradek | Renato, has there been anything on positive pinching vs positive curvature operator after the paper of Bourguignon and Karcher I linked to? | |
| Mar 17, 2017 at 22:28 | history | edited | Renato G. Bettiol | CC BY-SA 3.0 |
Added more info on the example of CP^n
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| Mar 17, 2017 at 22:23 | history | answered | Renato G. Bettiol | CC BY-SA 3.0 |