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.



Volume-preserving projective transformations are isometries
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.


