We will introduce the basics of the Projective Differential Geometry of geodesics, with particular attention to infinitesimal projective symmetries (i.e. vector fields whose local flow sends non-parametrized geodesics into non-parametrized geodesics), to projective equivalence (metrics admitting the same non-parametrized geodesics) and to the metrizability of projective structures including their relationship with integrable systems (in the sense of Liouville). The course will cover both classical theory and more recent research topics.

Basic notions on differential manifolds, tangent space, vector fields, Levi-Civita connection and geodesic curves.

After a brief recall of the prerequisites on differential manifolds, we will introduce the concept of projective connection (or projective structure) associated with an affine connection as an equivalence class of connections sharing the same geodesic curves. We will then introduce infinitesimal projective symmetries (i.e. vector fields whose local flow sends geodesic curves into geodesic curves) which will be understood as particular Lie symmetries of a certain system of ODEs. We will then focus on the problem of the metrizability of projective structures, which will be mainly treated in the case of geometries (i.e. metrics) having projective symmetries (Lie problem). The final part of the course will explain the relation between metrizability and (super-) integrability (in the sense of Liouville) of the geodesic flow.

On site

Oral presentation

P.D.1-2 - February