Timeline for Volume-preserving projective transformations are isometries
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Mar 23, 2014 at 11:22 | comment | added | Dan Fox | To finish: I agree with what I think Matveev is saying - that from a modern perspective this is an immediate one-line proof fact that requires no citation. It's much much simpler than many other things (e.g. uniformization) that we all use without citation. | |
| Mar 23, 2014 at 11:19 | comment | added | Dan Fox | A modern reference where the fact is stated explicitly (in terms of volume forms) is Proposition 4 of Nomizu and Pinkall's paper in Tohoku 39 (1987). Finally, my mention of Cap and Slovak was meant only to give a refence for Matveev's "help of deep mathematics". If one works out explicitly what are "Weyl structures" and "bundles of scales" for the parabolic geometries corresponding to projective structures one gets back to the statement that there is a unique torsion free representative of a projective class of connections inducing a given principal connection on the bundle of volume densities | |
| Mar 23, 2014 at 11:11 | comment | added | Dan Fox | To complete the comment: in these old references there is not usually explicit mention of volume forms (except in Weyl), rather they speak of choosing coordinate systems, and relate the trace of the Christoffel symbols to the Jacobian of the coordinate transformations. To me this seems a matter of language - at the time they mostly worked locally, and the language of tensors/forms/densities was not used in the modern way. | |
| Mar 23, 2014 at 11:05 | comment | added | Dan Fox | @Matveev: to continue my comment, an explicit reference where this fact is stated explicitly is the italicized lemma on p. 104 of Eisenhart's 1927 book Non-Riemannian Geometry. Actually Eisenhart speaks only of normalizing the trace of the Christoffel symbols by an appropriate choice of coordinates and not explicitly of volume forms, but this is equivalent, and the uniqueness is explicit. In a footnote Eisenhart references a 1926 paper of JM Thomas (TAMS v.28 n.4) where the same result is stated as Theorem 1. | |
| Mar 23, 2014 at 10:54 | comment | added | Dan Fox | @Matveev: It is hard to write precisely in these small comment boxes. My use of the word "first" was unfortunate and wrong; certainly you are right that the notion of projective equivalence is much older than Weyl. What maybe is not so much older is the explicit notion of projective equivalence of affine connections as such (maybe due to Weyl?). The paper of Weyl is one of the older ones where one finds the projective Weyl and Schouten tensors. Anyone writing these things down has to implicitly understand that a choice of volume fixes an affine connection representing the projective structure. | |
| Mar 21, 2014 at 16:39 | comment | added | Vladimir S Matveev | To Alvarez and @Dan Fox: the construction is definitly not mine, I have understood it speaking with people from parabolic geometry and these people indeed read Thomas very carefully. Contrary to what Dan is saying, I do not think that Weyl 1921 is a good reference since the construction is not there and which is not the first place projective equivalence was introduced (I suggest Beltrami 1864 or Levi-Civita 1896). And citing the wonderful book of Cap and Slovak in this context can be very misleading for a reader of your paper. Sorry for no help but may be Dan can give a precise referens | |
| Mar 21, 2014 at 16:08 | comment | added | Dan Fox | That a nonvanishing density of nontrivial weight is preserved by a unique torsion-free connection representing a given projective structure has been used since the 1920s. Probably if you look in H. Weyl's 1921 paper in which projective equivalence was introduced, it can be found there, at least implicitly. Certainly Veblen, Young, Thomas, etc. knew this fact explicitly. The general fact is that for a parabolic geometry modeled on $G/P$ with $G$ semisimple and $P$ parabolic there is a notion of Weyl structures parameterized by sections of a certain line bundle. See the book of Cap and Slovak. | |
| Mar 21, 2014 at 15:16 | comment | added | alvarezpaiva | We all have to cite results we can prove ourselves. The reference is not so much to the result as to the construction. Is it yours? | |
| Mar 21, 2014 at 15:13 | comment | added | Vladimir S Matveev | Unfortunately as a rule I do not know references for results that can be proven on one line. May be result stays in Eastwood's ``notes on projective geometry'' and if one wants one can explain this result with the help of deep mathematics but as I said it is easy to prove than to look for references | |
| Mar 21, 2014 at 14:40 | comment | added | alvarezpaiva | The "second question" was not a question, but the statement of a result. I like your local argument made global by the irreducibility of the metric. I guess that what you say about projective structures and volume forms can be found in the works of Veblen and Young, right? Not the projective geometry books, but their papers on projective connections. Is there a more modern reference? | |
| Mar 21, 2014 at 11:17 | history | answered | Vladimir S Matveev | CC BY-SA 3.0 |