A curvature obstruction to integrability

[摘要] 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 obstructthe integrability of almost-complex structures. It is shown that certain special complexstructures cannot coexist with non-flat constant curvature metrics, and a formal variationalrealization of these structures is provided. The approach followed here is direct, meaningthat it bypasses the classical theory. The idea is to find obstruction equations for theintegrability of almost-complex structures by way of Nijenhuis tensor derivatives. Thesenew equations involve the curvature of a torsion-free connection, and reveal the interplaybetween almost-complex and Riemannian geometries. Curvature scalars to detect noncomplexity in the compact case then arise in a natural way.

[学科分类] 工程和技术（综合）

[关键词] almost-complex structures;integrability;curvature