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What is a simple, elementary proof of the following result?

A continuously differentiable map from the unit sphere $S^n \subset \mathbb{R}^{n+1}$ $(n > 1)$ to itself that preserves volumes and sends great circles to great circles is an isometry.

I have a simple, but non-elementary proof of a more general result:

A continuously differentiable map from a Zoll Riemannian or reversible Finsler manifold to itself that preserves volumes and sends geodesics to geodesics is an isometry.

I don't think the generality makes the result more interesting (non-isometries sending geodesics to geodesics are not very many except in the case of real projective spaces and spheres) and I wonder if there is some easy, quick argument that takes care of the case of spheres with their canonical metrics.

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2 Answers 2

up vote 7 down vote accepted

As you must now given your title, there is a first classical argument that shows that (except for $n=1$, a case taking care of itself) if your map sends great circle to great circles, it must be projective, i.e. come from a linear map i.e. be a composition of a linear map with central projection to the sphere.

In particular, it also sends totally geodesic subspheres of every dimension to totally geodesic subspheres of the same dimension. Now, the volume assumption implies that it must preserve the angles between any two equators (since it preserves the volumes of the pieces cut out by these two equators). This shows that the map is conformal, and since it preserves the volume it must be an isometry.

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Oh, by the way: you do not need as much regularity as you assumed. –  Benoît Kloeckner Mar 21 at 12:25
    
... in particular, being a bijection is enough for the map to belong to $PGL(n+1,R)$ (for $n\ge 2$). –  Misha Mar 21 at 12:46
    
@BenoîtKloeckner: bien vu! Exactly the sort of elementary argument I was looking for. –  alvarezpaiva Mar 21 at 14:34
    
@Misha and Benoit: the title was for "connoisseurs", the statement of the result was to make it independent of previous acquaintance with projective geometry. Besides, in the general case I do need $C^1$. –  alvarezpaiva Mar 21 at 14:38

A pair (projective structure, volume form) allows one to canonically construct a torsion-free affine connection. This affine connection belongs to the projective structure and has the property that the volume form is parallel. It is an easy exercise to show its (local) existence; this affine structure is unique modulo multiplication by a constant. Thus, your map is affine for this affine connection (which is the Levi-Civita connection of the standard metric on the sphere by your assumptions) and since the metric of the sphere is irreducible you map must preserve the metric.

As you see the proof works for any Riemannian or pseudo-Riemannian metric and give also half of the answer on you second question. Moreover, it is a local proof and you do not need that the geodesics are closed. I did not think about the finslerian analog of this statement but would not be suprised if a similar statement holds in the finslerian case.

A related question is Projectively equivalent connections

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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? –  alvarezpaiva Mar 21 at 14:40
    
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 –  Vladimir S Matveev Mar 21 at 15:13
    
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? –  alvarezpaiva Mar 21 at 15:16
    
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. –  Dan Fox Mar 21 at 16:08
    
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 –  Vladimir S Matveev Mar 21 at 16:39

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