You better ask Victor about it -- he explained me the idea of the proof but I did not check the whole proof/construction. As far I remember the metric is smooth and quadratically convex.

to @Andrey Gogolev As far as I know the example itself was not discussed in the literature but the idea (which you and Misha Katz also explained in your answers) is definitely not new and the additional information came from study of Liouville metrics on the torus which is not a big deal and can be obtained by hands

to @alvarezpaiva Everybody wants to see this example of Bangert which is actually not an explicit example but a proof of the existence. The example is not real anylytic of course and is obtained as a limit of a sequence of finsler metrics

Of course. Sorry. As always I skipped the introduction and went to the question immediately. By the way, Victor Bangert claimed that he could proved the existence of a finsler metric on a closed surface of genus >1 whose geodesic flow is integrable -- he did not publish the proof as far as I know though

What is Morse-Hedlund theorem and why one can not make it better?

Juan-Carlos, projective Lichnerowicz-Obata conjecture says that on a closed Riemannian manifold of nonconstant sectional curvature every infintesimal projective symmetry is a Killing vector field. It therefore can not be applied to the standard sphere or to the standard real projective space because they do have constant sectional curvature

You would not love the initial proof which is bases on tensor calculations. It is quite short though and could be put on the 2-page Doklady paper which I do not have by hand and can not check therefore. Actually, the review in MathReviews already contain some information. If you want a conceptual proof, a symmetric space is einstein so the Thomas cone connection is actually the Levi-Civita connection of a certain metric on the cone over the manifold. Projectively related metrics correspond to parallel (0,2) tensors for this cone metric and because of many symmetries they do not exist

No, I dont do it in my answer. The result of Sinjukov is local, works in all signatures, and is about projectively equivalent metrics and not about projective transformations or connected groups of projective transformations. The arguments in my asnwer using Lichnerowciz-Obata relies on the existence of a big group of isometries of the metrics in your question. If there exists a (noninfinitesimal) projective transformation, then the pullback of the killing vector fields are projective vector fields and one can use Licherowicz-Obata. The Kähler result is also about projective equivalence.

Juan-Carlos, there were two mainstreams in the classical (<1990) theory of projectively equivalent metrics: the french'' (+ Lie) and japanese'' studied mostly infinitesimal projective transformations and ``soviet'' mostly projectively equivalent metrics. In the case your metric has a big group of symmetries, the results of both groups can be used by the following simple observation that I also explained in my answer: if the metrics are projectively equivalent, then an isometry of the first is projective transformation of the second.

I do not understand your comment. If you are asking whether in my answer I assumed that the projective transformation is actually an infinitesimal projective transformation, I did not do it and spoke about projectively equivalent metrics only. If you would like me to tell you more assuming the existence of an infinitesimal projective transformation, then globally (in the riemannian case) it is in arXiv:math/0407337 and locally in Solodovnikov 1956 in dimensions >2 and in arXiv:0802.2344 + arXiv:0705.3592 in dimension =2. In the case I misunderstood your comment, please explain

This is a comment on the comment of Mischa about "and the other way around is clearly faulse". The small correction of Anton that makes the argument correct is ``every totally geodesic two-dimensional submanifold of CP(n) is a projective line''.

The "Riemmanian" version of the question is not that trivial IMHO. A possible proof: the gauss curvature of the surface is constant because it is preserved by the isometries. It is positive since every compact surface must have a point where the curvature is positive. Then, the surface is diffeomorph to the sphere or to the projective plane. The latter can not be imbedded in $R^3$ (and the proof of this fact is much easier under the assumption that the curvature is positive) so the surface is the sphere. Now, by the Alexandrov theorem the sphere is standartly imbedded.