A curvature obstruction to integrability

TL;DR

This paper investigates how the curvature of Riemannian metrics can prevent certain almost-complex structures from being integrable, introducing new obstruction equations that connect curvature and complex structure integrability.

Contribution

It provides a direct method to derive curvature-based obstruction equations for integrability, bypassing classical $G$-structure theory, and reveals the interplay between curvature and complex structures.

Findings

Certain special complex structures cannot coexist with non-flat constant curvature metrics

New curvature obstruction equations involve derivatives of the Nijenhuis tensor

Curvature scalars can detect non-complexity in compact cases

Abstract

The classical theory of $G$-structures, which include almost-complex structures, explains the relationship between the curvature of compatible connections and integrability. This note is an effort to understand how the curvature of Riemannian metrics can obstruct the integrability of almost-complex structures. It is shown that certain special complex structures cannot coexist with non-flat constant curvature metrics, and a formal variational realization of these structures is provided. The approach followed here is direct, meaning that it bypasses the classical theory. The idea is to find obstruction equations for the integrability of almost-complex structures by way of Nijenhuis tensor derivatives. These new equations involve the curvature of a torsion-free connection, and reveal the interplay between the almost-complex and Riemannian geometries. Curvature scalars to detect non-complexity in the compact case then arise in a natural way.