Invariant operators on manifolds with almost Hermitian symmetric structures, II. Normal Cartan connections

TL;DR

This paper extends the theory of invariant operators on manifolds with almost Hermitian symmetric structures by establishing the existence and uniqueness of normal Cartan connections, including explicit formulas for these connections.

Contribution

It generalizes previous results to structures without torsion-free linear connections and provides explicit formulas for the canonical connections.

Findings

Proved existence and uniqueness of normal Cartan connections for a broader class of structures.

Derived explicit formulas for these connections in terms of underlying linear connection curvatures.

Extended classical results to first order structures with torsion.

Abstract

In the first part of this series of papers we developed the invariant differentiation with respect to a Cartan connection, we described this procedure in the terms of the underlying principal connections, and we discussed applications of this theory to the construction of natural operators. In this part we will extend the results of \cite{Ochiai, 70} on the existence and the uniqueness of the so called normal Cartan connections on manifolds with almost Hermitian symmetric structures to first order structures which do not admit a torsion free linear connection. Moreover, for each of these structures we obtain explicit (universal) formulae for these canonical connections in the terms of the curvatures of the underlying linear connections.