Publication details

A curvature obstruction to integrability

Authors

CLEMENTE Gabriella Alexandrea

Year of publication 2023
Type Article in Periodical
Magazine / Source Mathematical Communications
MU Faculty or unit

Faculty of Science

Citation
Web https://www.mathos.unios.hr/mc/index.php/mc/article/view/4572/875
Keywords almost-complex structures; integrability; curvature
Description 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 almost-complex and Riemannian geometries. Curvature scalars to detect non -complexity in the compact case then arise in a natural way.
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