Volume-preserving projective transformations are isometries 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.
 A: 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.
A: 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
