On the obstruction to integrability of almost-complex structures

TL;DR

This paper investigates the action of diffeomorphisms on the bundle of almost-complex structures, constructing a differential invariant that characterizes integrability via the Nijenhuis tensor.

Contribution

It introduces a new differential invariant of almost-complex structures and establishes its equivalence to the vanishing of the Nijenhuis tensor, linking invariance to integrability.

Findings

Constructed a nontrivial 1st order differential invariant.

Proved the invariant vanishes iff the Nijenhuis tensor is zero.

Connected the invariant's vanishing to integrability of almost-complex structures.

Abstract

The natural bundle $\pi:E\to M$ of almost-complex structures is considered. The action of the pseudogroup of all diffeomorphisms of $M$ on the total space $E$ is investigated. A nontrivial 1-st order differential invariant of this action is constructed. It is proved that the Nijenhuise tensor of an almost-complex structures is equal to zero iff the constructed invariant for this structure is zero.