Forschung / Research

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.

Historically, projective geometry appeared as a generalization 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 generalization of the theory of geodesically equivalent metrics and the generalization 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 hereExterner Link and the metrisability of projective structures interesting for general relativity were studied hereExterner Link. 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,  R. Gover and K. Neusser.

C-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 c-projective equivalence and h-projective transformations as an object to study. The role of geodesics in the c-projective setting is now played by thec-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 c-planar curves coincide. A diffeomorphism of a Kähler manifold is called h-planar if it mapsc-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 c-projectively equivalent metrics. A good example is a  proof of the Riemannian Yano-Obata-conjectureExterner Link which is an c-projective analog of the  projective Lichnerowicz-Obata conjectureExterner Link 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.

There similarity was completely understood in and used in  the proof of   Yano-Obata-conjecture in any signature Externer Link.

An approach to c-projective geometry from the viewpoint of Cartan geometry is developed in the book C-projective geometry.

In the framework of this subject we collaborated with A. Bolsinov, D. Calderbank, V. Kiosak, T. Mettler, K. Neusser and . S. Rosemann,

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 hereExterner Link and hereExterner Link.

Two survey papers containing in particular open problems in this topic are Open problems and questions about geodesics   and Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems.

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

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 hereExterner Link and hereExterner Link.

Generally, we are  trying to apply  methods coming from other branches of  geometry  to problems in Finsler geometry.  In particular, applying  methods coming from symplectic geometry we described Geodesic behavior for Finsler metrics of constant positive flag curvature on $S^2$.   Applying methods from convex geometry we solved   Laugwitz Conjecture and Landsberg Unicorn Conjecture for Minkowski norms with SO(k)×SO(n-k)-symmetry .   Combining  methods of Finsler geometry with that of stochastic processes we described Geodesic random walks, diffusion processes and Brownian motion on Finsler manifolds .

Within this project we collaborate with R. Bryant, M. Troyanov, I. Pavlyukevich and     Ming Xu.

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 hereExterner Link and hereExterner Link.

Certain members participated  in the Research training group "Quantum- and Gravitational fields"Externer Link. This group is for mathematicians interested 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, see hereExterner Link and hereExterner Link.

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

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 hereExterner Link.

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), Konrad  Schöbel  and Alexaner P. Veselov (Loughborough University).

The ultimate goal of this project is to start and develop a new area of research, Nijenhuis Geometry, with hundreds of mathematicians involved, big conferences every year, and many applications in different subjects of mathematics and mathematical physics.

Nijenhuis Geometry studies Nijenhuis operators, i.e., fields of endomorphisms on a smooth manifold such that their Nijenhuis torsion vanishes. Our strategy is to re-direct the research agenda in this area from local analysis at generic points to studying singularities and global properties, as it already happened with other areas of geometry. We will do it by answering the following natural questions:

(A) Local description: to what ‘normal’ form can one bring a Nijenhuis operator near almost every point by a local coordinate change?

(B) Singular points: what does it mean for a point to be generic or singular in the context of Nijenhuis Geometry? What singularities are typical? Non-degenerate? Stable? How do Nijenhuis operators behave near typical, non-degenerate and stable singular points?

(C) Global properties: what restrictions on a Nijenhuis operator are imposed by compactness of the manifold? What are topological obstructions for a manifold carrying a Nijenhuis operator with specific properties (e.g. with singular points of prescribed type)?

Vanishing of Nijenhuis torsion is the simplest differential-geometric condition on a field of endomorphisms and that is the reason why Nijenhuis operators appear independently in many unrelated problems of differential geometry and mathematical physics. Results on Nijenhuis operators obtained within this project will imminently be applied in these topics. We envisage applications in finite-dimensional integrable systems, in the theory of geodesically equivalent metrics, in integrable infinite dimensional systems of hydrodynamic type and in classical differential geometry.

The program  paper on this topic is  Nijenhuis Geometry;  further results demonstrating that the programme  is  realistic are Nijenhuis Geometry III: GL-regular Nijenhuis operators,  Applications of Nijenhuis geometry: Nondegenerate singular points of Poisson-Nijenhuis structures , Applications of Nijenhuis geometry II: maximal pencils of multihamiltonian structures of hydrodynamic type,  Applications of Nijenhuis geometry III: Frobenius pencils and compatible non-homogeneous Poisson structures.

Within this project we collaborate with  A. Konyaev and A. Bolsinov.