Tuesday 10 May 2022
10:00-10:50 Thomas Mettler
Deformations of the Veronese embedding and Finsler 2-spheres of constant curvature
Abstract:
A path geometry on a surface M prescribes a path for each direction in every tangent space. A path geometry may be encoded in terms of a line bundle P on the projectivised tangent bundle P(TM) of M. Besides P, the projectivised tangent bundle is also equipped with the vertical bundle L of the base-point projection P(TM) -> M. Interchanging the role of L and P leads to the notion of duality for path geometries. In my talk I will discuss joint work with Christian Lange (Munich), where we investigate global aspects of the notion of duality for Finsler 2-spheres of constant curvature and with all geodesics closed. In particular, we construct new examples of such Finsler 2-spheres from suitable deformations of the Veronese embedding.
11:00-11:50 The Joachimsthal constant on (asymmetric) hyperquadrics
Sergio Benenti
12:00-14:00 Lunch Break
14:00-14:40 Claudia Chanu
L-tensors without real simple eigenvalues: some examples and possible applications in order to solve geodesic Hamilton-Jacobi equation.
Abstract:
L-tensors, also known as Benenti tensors, are special conformal Killing tensors on Riemannian manifolds allowing the construction of a chain of Poisson commuting independent first integrals of the geodesic flow with common normal eigenvectors. The coordinates adapted to the eigenvector frame allow the determination of a complete integral of the Hamilton-Jacobi equation for the two projective equivalent geodesic Hamiltonians. In Benenti's approach the fundamental assumption was that eigenvalues were real and simple. We are trying to see what can be said about the solution of the Hamilton Jacobi equation when the fundamental condition is not satisfied.
15:00-15:40 Andreas Vollmer
(Pseudo-)Riemannian Metrics with projective symmetries and their superintegrability
Abstract:
Vector fields whose local flow preserves geodesics up to reparametrisation ("pre-geodesics") are called projective vector fields. The description of (pseudo-)Riemannian metrics admitting projective vector fields is a classical problem: It was formulated, for dimension 2, by Sophus Lie in the 1880s, and has been solved in recent years, in terms of explicit local normal forms. The higher dimensional problem has been studied by Aleksandr Solodovnikov (1956) who showed that the problem reduces to a system of ODEs for a certain class of n-dimensional Levi-Civita metrics.
The talk will outline the known result for Lie's classical problem, and also present an explicit description of (all) Levi-Civita metrics in dimension 3.
In a second part, the talk will address the integrability (in the sense of Liouville and Arnol'd) of Hamiltonian systems associated with metrics admitting projective vector fields, particularly their superintegrability in dimension 2 (and 3).
16:00-16:40 Jan Schumm
C-projective equivalence and quantum-integrability, Kähler metrics with c-projective vector fields
Abstract:
a) C-projective equivalence is the Kähler analogue of projective equivalence: J-planar curves are a generalization of geodesics on Kähler manifolds and two metrics are called c-projectively equivalent if they have the same (unparametrised) J-planar curves. Analogously to projective geometry the existence of a nonproportional c-projectively equivalent metric conveys commuting integrals of the geodesic flow. In projective geometry the associated differential operators commute. We show that also in the c-projective case the differential operators associated with the integrals commute.
b) A projective vector field is characterized by its local flow mapping geodesics to geodesics. In 1882 Sophus Lie asked to describe all metrics that admit such vector fields. The Kähler analogue is a c-projective vector field which maps J-planar curves to J-planar curves. We classify metrics admitting c-projective vector fields and compute normal forms.
17:00-18:00 Open Discussion
19:30 Dinner