We investigate existence and uniqueness of minimal Lagrangian connections on smooth projective manifolds as well as properties of related dynamical systems.
A projective structure on a smooth manifold consists of an equivalence class p of torsion-free connections on its tangent bundle, where two such connections are called equivalent if they have the same geodesics up to parametrisation.
The representative connections of a projective structure p on a smooth manifold M are in one-to-one correspondence with the sections of an affine bundle, whose total space carries a split-signature metric as well as a symplectic form, both of which are defined in a canonical fashion from p. Consequently, all the submanifold notions of (pseudo-)Riemannian geometry and symplectic geometry can be applied to the representative connections of p. This point of view gives rise to the notion of a minimal Lagrangian connection.
The aim of this research project is to investigate various questions related to minimal Lagrangian connections, in particular, to provide a characterisation of projective manifolds that arise from a minimal Lagrangian connection.
