Plan of Research

In the last years I am mostly trying to understand when two different metrics on one manifold can have the same geodesics. Now I more-or-less understand the Riemannian case, and try to understand the case when both metrics are pseudo-Riemannian.

Two pseudo-Riemannian metrics on one manifold are called geodesically equivalent, if their geodesics coincide as unparameterised curves.

My ultimate goal is to understand completely the geometry and topology of closed manifolds admitting geodesically equivalent metrics. The first steps in achieving this goal could be as follows.

To solve the Beltrami Problem (1865) for closed pseudo-Riemannian 2-manifolds:
To describe all nonproportional geodesically equivalent metrics,
to prove or disprove in the pseudo-Riemannian case the Projective Lichnerowicz-Obata Conjecture:
Let (M,g) be a closed connected pseudo-Riemannian manifold of dimension at least two. Suppose a connected Lie group acts on the manifold by projective transformations. Then it acts by isometries, or g has constant sectional curvature,
to find first obstructions preventing a closed manifold from possessing geodesically equivalent pseudo-Riemannian metrics.

I am attacking pseudo-Riemannian case because in the last four years I solve all these problems in the Riemannian case. The Riemannian analog of the Beltrami problem for closed 2-manifolds was understood already 1998 (this is a joint work with P. Topalov). The only technical trick we used came from the theory of integrable geodesic flows (we mostly used results of Kolokoltsov). One can find the proof in the following extended version of my joint paper with A. V. Bolsinov and A. T. Fomenko Riemannian metrics with integrable geodesic flows on surfaces: local and global geometry,

The proof of Riemannian Lichnerowicz-Obata conjecture is much more complicated. It is written in Projective Lichnerowicz-Obata Conjecture .

Now I also completely understand the topology of closed manifolds admitting Riemannian geodesically equivalent metrics:

Geodesic Rigidity Theorem: Let M be closed connected. Let two Riemannian metrics on M be geodesically equivalent and nonproportional. Then the manifold can be covered by the sphere, or it admits a metric with reducible holonomy group .

Unfortunately, the proof of the Geodesic Rigidity Theorem is not written jet. It is going to be huge (the parts that are already written need around 40 pages).

The success of my approach (as I told before, certain parts of this approach were developed in close cooperation with other people. For example, as I mentioned before, the first steps were done together with Topalov) suggests that one should try to generalise the methods which allowed me to understand the Riemannian case to pseudo-Riemannian metrics.

Click here to read more about the methods I used in the Riemannian case, and about the expected difficulties of the pseudo-Riemannian case. Click here to know whether I think it is possible to do the whole program.