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Geodesically equivalent metrics and projective transformations

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 problem and second problem, and the classical projective Lichnerowicz-Obata conjecture. 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 here and here.

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

Projective geometry

Historically, projective geometry appeared as a generalisation of and a tool in the theory of geodesically equivalent metrics. The basic geometric structure there is again the set of (unparametrised) geodesics of a symmetric affine connection. Two connections are said to be projectively equivalent if geodesics of the first are reparametrized geodesics of the second. This generalisation allows one to effectively apply the ideas of Tracy Thomas and Elie Cartan on the geometrization of ordinary and partial differential equations - nowadays, projective geometry is one of the most popular Cartan geometries.

As we mentioned above, projective geometry is a generalisation of the theory of geodesically equivalent metrics and the generalisation is not a tautological one: it appears that most projective structures are not metrizable in the sense that for most symmetric affine connections, there is no metric such that its Levi-Civita connection is projectively equivalent to the given connection.  The question whether a given projective structure is metrizable and, if so, how big is the set of metrics within this projective structure, is one of the questions we have studied. The equations governing the problem were obtained here and the metrisability of projective structures interesting for general relativity were studied here. Moreover, the metrisablity problem was very essential in the solution of the above mentioned second problem of Lie.

In this project, we collaborate with M. Eastwood and R. Gover.

H-projective geometry

H-projective geometry was introduced as an analog of projective geometry on complex (or almost complex) manifolds in the 1950's by Otsuki and Tashiro. They observed that there are only trivial examples of pairs of geodesically equivalent Kähler metrics, so projective geometry is not that interesting in the Kähler setting. Instead of geodesic equivalence and projective transformations, they suggested the h-projective equivalence and h-projective transformations as an object to study. The role of geodesics in the h-projective setting is now played by the h-planar curves. These curves are defined by the condition that the acceleration is complex proportional to the velocity in each point of the curve. Two Kähler metrics are called h-projectively equivalent if their h-planar curves coincide. A diffeomorphism of a Kähler manifold is called h-planar if it maps h-planar curves to h-planar curves.

The similarity with the theory of geodesically equivalent metrics has the following advantage: most problems stated or solved by classics in the theory of geodesically equivalent metrics have analogs in the theory of h-projectively equivalent Kähler metrics. Moreover, sometimes the analogs of methods that worked for geodesically equivalent metrics still work for h-projectively equivalent metrics. A good example is the recent proof of the Yano-Obata-conjecture which is an h-projective analog of the  projective Lichnerowicz-Obata conjecture mentioned above. This is not the only example and the theory of h-projectively equivalent metrics will stay in our focus for the next few years.

In the framework of this subject we collaborated with D. Calderbank, V. Kiosak and T. Mettler.

Finite dimensional integrable Hamiltonian systems

We mostly work with "natural" Hamiltonian systems, i.e., systems that live on the cotangent bundle with hamiltonian given as the sum of kinetic (i.e. quadratic in the momenta) and potential energy. One of the motivations for our interest is that they are a powerful tool in the study of geodesically equivalent metrics and projective transformations. Indeed, they were intensively used in the proof of the Lichnerowicz-Obata conjecture and the Yano-Obata-conjecture mentioned above and also in the description of the topology of manifolds admitting geodesically equivalent metrics.

Besides this application of finite dimensional integrable systems in projective and h-projective geometry, we are trying to understand what metrics on closed two-manifolds admit integrals that are polynomial in momenta. The topology of these manifolds was already understood by Kolokoltsov: they must have vanishing Euler characteristic. Also the cases when the degree of the integral is one or two are well understood. The next natural case is when the degree of the integral is three. In this case, almost nothing is known and there are very few examples on closed surfaces. Two of these examples were constructed by members of our group, see here and here.

Within this project, we collaborate in particular with H. Dullin, B. Kruglikov and V. Shevchishin.

Finsler geometry

Finsler geometry is a classical generalisation of Riemannian geometry. The role of the Riemannian metric plays a function F on the tangent bundle such that its restriction to each tangent space is a Minkowski norm. Our recent results in this theory are due to the observation that, given a Finsler metric, one can canonically construct a Riemannian metric that induces the Finsler metric. This construction appeared to be very effective and with the help of it, we solved a bunch of named problems, see here and here.

Within this project we collaborate with M. Troyanov.

General Relativity and Mathematical Physics

We do not really research in the field of General Relativity or Mathematical Physics. However, when we obtain a result we always try to look for possible applications in physics. Sometimes and not surprisingly, this is possible since both subjects, differential geometry and the theory of integrable systems appeared as an attempt to create mathematical tools for physics.

Some problems in the theory of geodesically equivalent metrics were actually stated by physists. One of such problems was formulated in different versions by H. Weyl, Z. Petrov and J. Ehlers and was solved here and here.

Certain members participate in the Research training group "Quantum- and Gravitational fields". This group is for mathematicians intertested in physics and for physists that need mathematics in their work. Our project within this group is the study of the integrability of the geodesic flows of metrics interesting for general relativity and quantum field theory. Recent results in this direction could be found here and here.

Even the Finsler geometry may have relation to General Relativity. Our contribution to this topic is here.

We are also trying to understand whether the integrability of the geodesic flow could be generalised up to the quantum integrability. For the integrals appearing in the framework of geodesic equivalence this "quantisation" was obtained here.

An algebraic approach to the Separation of Variables

Separation of Variables is one of the most important techniques for solving partial differential equations and it is a long standing problem to classify all coordinate system in which a classical equation such as the Laplace, the Hamilton-Jacobi or the Schrödinger equation can be solved in this way - the so called separation coordinates.

Separation of Variables is related to Finite Dimensional Integrable Hamiltonian Systems, because every system of separation coordinates gives rise to a completely integrable Hamiltonian system. It is also related to Projective Geometry, since any geodesically equivalent metric defines a system of separation coordinates. However, not all separation coordinates arise in this way and it is not clear to which extent, or in which sense, separation of variables is a projectively invariant concept.

In order to completely understand these relations for the most important class of manifolds, we developed a purely algebraic approach to the separation of variables for (pseudo-)Riemannian manifolds of constant sectional curvature, including Euclidean and Minkowski space. This takes the problem of Separation of Variables to the realm of Representation Theory, Algebraic Geometry and Geometric Invariant Theory. Using this approach, we were able to show, for example, that the moduli space of separation coordinates on a sphere is isomorphic to a celebrated object in Algebraic Geometry - the moduli space of stable algebraic curves of genus zero with marked points.

We currently apply our approach to all constant curvature manifolds. With this knowledge our aim is to eventually extend it to manifolds of non-constant curvature by using methods from Projective Geometry.

In this project we collaborate with Robert Milson (Dalhousie University Halifax) and Alexaner P. Veselov (Loughborough University).