Linear Connections and Almost Complex Structures

Abstract

An almost complex structure is defined on $P$, the principal bundle of linear frames over an arbitrary even-dimensional smooth manifold $M$ with a given linear connection. Complexifying connections are those which induce a complex structure on $P$. For two-dimensional manifolds, every linear connection is of this kind. In the special case where $M$ itself is an almost complex manifold, a relationship between the two almost complex structures is found and provides a very simple proof of the fact that the existence of an almost complex connection without torsion implies the integrability of the given almost complex structure. As a second application, we give a geometrical interpretation of an identity between the torsion of an almost complex structure on $M$ and the torsion of an almost complex connection over $M$.