Timeline for 'Unitary' charts on odd-dimensional spheres
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Mar 21, 2015 at 23:14 | vote | accept | David Roberts♦ | ||
| Mar 21, 2015 at 23:14 | comment | added | David Roberts♦ | Thanks for the expansion of part II! This is the part I needed: "the induced isomorphism between the corresponding holomorphic tangent spaces is always unitary up to a scalar multiple." | |
| Mar 21, 2015 at 11:49 | history | edited | Robert Bryant | CC BY-SA 3.0 |
added some more detail on the isomorphism
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| Mar 20, 2015 at 12:00 | history | edited | Robert Bryant | CC BY-SA 3.0 |
added 1397 characters in body
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| Mar 19, 2015 at 21:06 | comment | added | David Roberts♦ | Ah, my apologies for being vague! "comes from" was meant to be a bit more general than my original question about unitarity, which as you say is false. Rescaling is perfectly ok, if necessary! In fact I expected some rescaling, since that's what I found in the conformal case... Thanks for your patience in dragging this out of me :-) | |
| Mar 19, 2015 at 11:48 | comment | added | Robert Bryant | @DavidRoberts: I think that what may be missing is clarification of the ill-defined 'comes from' in the above statement. When considering, say, the inverse of stereographic projection $\sigma$, which is a conformal diffeomorphism from $\mathbb{R}^n$ to the complement of a point in $S^n$, one finds an orthonormal tangent frame field by letting $e_i$ be $\sigma'(\partial/\partial x^i)$ after it has been normalized to have unit length. But you seem to be trying to get a CR-unitary frame field directly from a chart, without normalizing, and this is not possible. | |
| Mar 19, 2015 at 9:54 | comment | added | David Roberts♦ | (You know the following well, I imagine, but let me write it out for my own clarity of thinking) One can think of points in $S^{2n+1}\subset \mathbb{C}^{n+1}$ as the last column of a matrix in $SU(n+1)$, and an appropriate basis of $HS^{2n+1}$, considered as vectors in the ambient $\mathbb{C}^{n+1}$, supplies the other $n$ columns. Ideally this basis comes from a complex analogue of a conformal map that supplies a chart, as happens when thinking of the round sphere and local sections of $O(n+1) \to S^n$. Is there any particular obstruction to getting such a frame field from a coordinate chart? | |
| Mar 19, 2015 at 8:53 | comment | added | Robert Bryant | @DavidRoberts: I'm mystified by your comment above, since I don't really know where this 'chart' is supposed to take $U$. Also, your final comment in the question, about the chart being equivalent to finding a section of the bundle $\mathrm{SU}(n{+}1)\to S^{2n+1}$ over $U$, doesn't make sense to me either. Such a section, aka a local frame field along $U$, certainly does exist, but it's not going to be the framing associated to any coordinate chart. | |
| Mar 19, 2015 at 6:01 | comment | added | David Roberts♦ | Hmm, not really the answer I wanted to hear, since that rules out the CR chart I was looking at (but at least it stops me from wrestling with futile calculations). My main question still stands, namely: what is some chart that is unitary in the way described in the post? | |
| Mar 19, 2015 at 1:37 | history | answered | Robert Bryant | CC BY-SA 3.0 |