Cone structures and conic connections

A cone structure on a manifold $M$ is given by a closed submanifold $\mathcal C\subset \mathbb P TM$ of the projectived tangent bundle of $M$, which is submersive over $M$. Such geometric structures arise naturally in differential and algebraic geometry and they come often equipped with a conic connection, which specifies a distinguished family of curves on $M$ in directions of $\mathcal C$. In a joint work with Jun-Muk Hwang we defined an important local invariant for so-called characteristic conic connections, namely the cubic torsion. In this talk I will give a geometric interpretation of the cubic torsion and will discuss some applications. This talk is based on joint work in progress with Andreas Čap.