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We construct a family of canonical connections and surrounding basic theoryfor almost complex manifolds that are equipped with an affine connection. Thisframework provides a uniform approach to treating a range of geometries. Inparticular we are able to construct an invariant and efficient calculus forconformal almost Hermitian geometries, and also for almost complex structuresthat are equipped with a projective structure. In the latter case we find aprojectively invariant tensor the vanishing of which is necessary andsufficient for the existence of an almost complex connection compatible withthe path structure. In both the conformal and projective setting we givetorsion characterisations of the canonical connections and introduce certaininteresting higher order invariants.

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