Timeline for 'Unitary' charts on odd-dimensional spheres
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 21, 2015 at 23:14 | vote | accept | David Roberts♦ | ||
| Mar 19, 2015 at 1:37 | answer | added | Robert Bryant | timeline score: 5 | |
| Mar 18, 2015 at 22:58 | history | edited | David Roberts♦ | CC BY-SA 3.0 |
Clarified remark about CR isomorphism
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| Mar 18, 2015 at 22:56 | comment | added | David Roberts♦ | @RobertBryant my fault: I didn't mean $\mathbb{C}^n\times\mathbb{R}$ with its usual CR manifold structure, see my comment to Benoît. | |
| Mar 18, 2015 at 22:54 | comment | added | David Roberts♦ | @BenoîtKloeckner Yes, I meant $\mathbb{C}^n\times \mathbb{R}$ in its guise as the Heisenberg group (I'm afraid I'm used to thinking of diffeomorphisms, not worrying about metric or complex structures). I'm not entirely sure that particular chart is going to help, which is why I'm asking here. | |
| Mar 18, 2015 at 10:56 | comment | added | Benoît Kloeckner | Just as Robert Bryant, I am puzzled by your assertion that $\mathbb{C}^n\times \mathbb{R}$ be CR-isomorphic to any open set of the sphere: to phrase Robert's objection differently, they live in opposite parts of the CR world, the former being Levy-flat and the later being strictly pseudoconvex. The relevant keyword is probably "Heisenberg group": in one of its guises, it is a pseudo-Riemannian manifold that parametrizes any complement of a point in the unit sphere of $\mathbb{C}^n$, and it can certainly be endowed with a natural CR-structure. | |
| Mar 18, 2015 at 10:42 | comment | added | Robert Bryant | I am not sure what you mean by a 'CR isomorphism'. There is no diffeomorphism $\phi:\mathbb{C}^n\times\mathbb{R}\to U$ such that the differential of $\phi$ is complex linear on the tangent vectors to the complex hypersurfaces $\mathbb{C}^n\times\{x\}$, which is what I would think is meant by 'CR isomorphism' in this case. (If such a $\phi$ existed, then $\phi\bigl(\mathbb{C}^n\times\{x\}\bigr)\subset U\subset S^{2n+1}$ would be a complex submanifold of $\mathbb{C}^{n+1}$, which, by the maximum principle, is impossible.) | |
| Mar 18, 2015 at 4:38 | comment | added | David Roberts♦ | Notice that I really do want $U$, and not some other smaller open subspace! | |
| Mar 18, 2015 at 4:36 | history | asked | David Roberts♦ | CC BY-SA 3.0 |