Geometry of conic connections

Abstract

A cone structure on a complex manifold $M$ is a closed submanifold $\mathcal C\subset \mathbb P TM$ of the projectivized tangent bundle of $M$ that is submersive over $M$. So this defines a set $\mathcal C_x$ of distinguished directions in each point $x\in M$. A conic connection on $\mathcal C$ is then a family of unparametrized curves on $M$ that comprises exactly one curve through each point $x\in M$ in each direction in $\mathcal C_x\subset \mathbb PT_xM$. This can be encoded by a line subbundle $\mathcal F\subset T\mathcal C$. The subclass of characteristic conic connections is defined by the vanishing of a simple invariant, called characteristic torsion. For those, one has a much more subtle and slightly mysterious invariant called the cubic torsion. The first aim of this article is to provide a new approach to the cubic torsion, which also leads to a geometric condition characterizing its vanishing. We then specialize to the case of isotrivial cone structures, for which the fibers of $\mathcal C$ are assumed to be of some fixed type. Such a structure induces a first-order $G$-structure on $M$ whose structure group is the projective automorphism group of the model fiber. Moreover, any connection $γ$ on the associated $G$-structure induces a conic connection $\mathcal F^γ$ on $\mathcal C$. Assuming that the model fiber is homogeneous, we study the relation between the torsion and curvature of a connection $γ$ and the characteristic and cubic torsion of $\mathcal F^γ$. As an application we discuss cone structures of subadjoint type, showing in particular that there are such structures admitting conic connections with vanishing characteristic and cubic torsion that are not locally flat.