Recall that two pseudo-Riemannian metrics g and g' are called geodesically equivalent if every g'-geodesic, considered as an unparametrized curve, is a g-geodesic. A diffeomorphism of a Riemannian manifold is called projective transformation if it takes (unparametrized) geodesics to geodesics. These are very classical subjects. The first examples of non-proportional geodesically equivalent metrics are due to Lagrange 1789. At the end of the 19th century, geodesically equivalent metrics were one of the favorite research topics. The classics (Dini, Lie, Liouville, Levi-Civita, Weyl) did a lot but not everything. Remarkably, they left a list of problems they considered but could not solve.

The theory of geodesically equivalent metrics and projective transformations was one of the main research topics of our research group in the past decade and we managed to solve some of the classical problems including two problems of Sophus Lie, see first problemExterner Link and second problemExterner Link, and the classical projective Lichnerowicz-Obata conjectureExterner Link. We also understood completely the topology of a closed manifold admitting geodesically equivalent Riemannian metrics. The proof of this result is not written yet but the preliminaries can be found hereExterner Link and hereExterner Link.

In the framework of this theory, we collaborated with many mathematicians, in particular with R. Bryant, A. Bolsinov and V. Kiosak.