Calculus and invariants on almost complex manifolds, including projective and conformal geometry

TL;DR

This paper develops a unified framework for almost complex manifolds with affine connections, introducing invariants and canonical connections for conformal and projective geometries, and characterizing their torsion and higher order invariants.

Contribution

It constructs a family of canonical connections and invariants for almost complex manifolds with affine structures, unifying conformal and projective geometries.

Findings

Invariant calculus for conformal almost Hermitian geometries

Projectively invariant tensor characterizing compatible connections

Torsion characterizations and higher order invariants for canonical connections

Abstract

We construct a family of canonical connections and surrounding basic theory for almost complex manifolds that are equipped with an affine connection. This framework provides a uniform approach to treating a range of geometries. In particular we are able to construct an invariant and efficient calculus for conformal almost Hermitian geometries, and also for almost complex structures that are equipped with a projective structure. In the latter case we find a projectively invariant tensor the vanishing of which is necessary and sufficient for the existence of an almost complex connection compatible with the path structure. In both the conformal and projective setting we give torsion characterisations of the canonical connections and introduce certain interesting higher order invariants.