The project aims at investigating geodesics of Riemannian and semi-Riemannian metrics. We wish to achieve two goals: bound chaoticity/complexity of the geodesic flow and understand when it can be described completely. The latter means integrability of the equations and we wish to investigate existence of polynomial in velocities integrals for general metrics/respectively metrics with special properties, most importantly Einstein metrics.
The methods to study this problem are related to geometric theory o f overdetermined systems of partial differential equations. A group in Tromsø has experts in this area. In particular, Kruglikov has successively applied these methods to old classical problem of invariant characterisation of two-dimensional metrics with quadratic integrals.
On the other hand, the group in Jena investigates geodesic flows for Riemannian and pseudo-Riemannian metrics. Matveev is a world expert in the theory of geodesic equivalence. His methods coupled with geometric ideas from tensor and tractor calculus will facilitate application of the general methods of differential equations mentioned above.
We would like to indicate that a particular important case of the considered problem constitute Lorenzian metrics of signature (1,3). In this c ase the constraint is the Einstein field equations and the geodesics correspond to particle trajectories, e.g. null-geodesics for the light. Thus the integrability problem is of both mathematical and physical interest.