The course will give an introduction to the basics of Projective Differential Geometry. In particular, it will cover infinitesimal projective symmetries (i.e. vector fields whose local flow preserves unparametrized geodesics), projective equivalence of pseudo-Riemannian metrics (equivalent metrics share the same unparametrized geodesics), and the metrizability of projective structures including their relationship with integrable systems (in the sense of Liouville). The course will cover both the classical theory and more recent research findings.

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After a brief recapitulation of prerequisites like differentiable manifolds, connections & geodesics, the course will set out to introduce the projective equivalence of affine connections, leading to projective structures whose properties we discuss. We then continue with infinitesimal projective symmetries (i.e. vector fields whose local flow preserves unparametrized geodesics): they can be regarded as particular symmetries of a certain system of ODEs. With these notions at hand, the course will proceed to the the problem of metrizability of projective structures, which will touch geometries (i.e. metrics) with projective symmetries (known as Lie's Problem). Various techniques that have been applied to arrive at normal forms of such metrics will be presented. The course will end with a discussion of the link between metrizability and (super-)integrability (in Liouville's sense).

Mixed mode

Oral presentation

P.D.2-2 - April