Vladimir S. Matveev

Geometrical interpretation of Benenti systems

Abstract We show that the following two separately developed theories, the theory of Benenti systems in mathematical physics and the theory of projectively equivalent metrics in classical differential geometry, study essentially the same object. Combining methods and results from these two theories, one can prove the commutative integrability of projectively equivalent pseudo-Riemannian metrics and construct infinitely many new Hamiltonian systems, integrable in the classical and in the quantum sense.

Geodesic Equivalence via Integrability

We suggest a construction that, given an orbital diffeomorphism between two Hamiltonian systems, produces integrals of them. We treat geodesic equivalence of metrics as the main example of it. In this case, the integrals commute; they are functionally independent if the eigenvalues of the tensor giαg¯αj are all different; if the eigenvalues are all different at least at one point then they are all different at almost each point and the geodesic flows of the metrics are Liouville integrable. This gives us topological obstacles to geodesic equivalence.

Metric Connections in Projective Differential Geometry

We search for Riemannian metrics whose Levi-Civita connection belongs to a given projective class. Following Sinjukov and Mikes, we show that such metrics correspond precisely to suitably positive solutions of a certain projectively invariant finite-type linear system of partial differential equations. Prolonging this system, we may reformulate these equations as defining covariant constant sections of a certain vector bundle with connection. This vector bundle and its connection are derived from the Cartan connection of the underlying projective structure.

A solution of a problem of Sophus Lie: normal forms of two-dimensional metrics admitting two projective vector fields

We give a complete list of normal forms for the two-dimensional metrics that admit a transitive Lie pseudogroup of geodesic-preserving transformations and we show that these normal forms are mutually non-isometric. This solves a problem posed by Sophus Lie.

The Binet–Legendre Metric in Finsler Geometry

For every Finsler metric F we associate a Riemannian metric gF (called the Binet‐ Legendre metric). The Riemannian metric gF behaves nicely under conformal deformation of the Finsler metric F , which makes it a powerful tool in Finsler geometry. We illustrate that by solving a number of named Finslerian geometric problems. We also generalize and give new and shorter proofs of a number of known results. In particular we answer a question of M Matsumoto about local conformal mapping between two Minkowski spaces, we describe all possible conformal self maps and all self similarities on a Finsler manifold. We also classify all compact conformally flat Finsler manifolds, we solve a conjecture of S Deng and Z Hou on the Berwaldian character of locally symmetric Finsler spaces, and extend a classic result by H C Wang about the maximal dimension of the isometry groups of Finsler manifolds to manifolds of all dimensions. Most proofs in this paper go along the following scheme: using the correspondence F 7! gF we reduce the Finslerian problem to a similar problem for the Binet‐ Legendre metric, which is easier and is already solved in most cases we consider. The solution of the Riemannian problem provides us with the additional information that helps to solve the initial Finslerian problem. Our methods apply even in the absence of the strong convexity assumption usually assumed in Finsler geometry. The smoothness hypothesis can also be replaced by a weaker partial smoothness, a notion we introduce in the paper. Our results apply therefore to a vast class of Finsler metrics not usually considered in the Finsler literature.

Complete Einstein Metrics are Geodesically Rigid

We prove that every complete Einstein (Riemannian or pseudo-Riemannian) metric g of nonconstant curvature is geodesically rigid: if any other complete metric

Proof of the Projective Lichnerowicz Conjecture for Pseudo-Riemannian Metrics with Degree of Mobility Greater than Two

{{\bar{g}}}

Three-dimensional manifolds having metrics with the same geodesics

has the same (unparametrized) geodesics with g, then the Levi-Civita connections of g and