On the space of almost complex structures

TL;DR

The paper investigates the geometric structure of the space of almost complex structures on manifolds, revealing it as an infinite-dimensional complex pseudo-Riemannian manifold and analyzing its curvature properties.

Contribution

It introduces a natural parametrization of the space of almost complex structures and studies its geometric and curvature properties, including special cases on symplectic and Riemannian manifolds.

Findings

The space of almost complex structures is an infinite-dimensional complex weak pseudo-Riemannian manifold.

The curvature of this space is explicitly computed.

Special cases include the spaces of associated and orthogonal almost complex structures.

Abstract

The space ${\mathcal A}$ of almost complex structures on a closed manifold $M$ is studied. A natural parametrization of the space ${\mathcal A}$ is defined. It is shown, that ${\mathcal A}$ is a infinite dimensional complex weak Pseudo-Riemannian manifold. A curvature of the space ${\mathcal A}$ is found. The space ${\mathcal A}_\omega$ of associated almost complex structures on a symplectic manifold $M,\omega$ and space ${\mathcal AO}$ of orthogonal almost complex structures on a Riemannian manifold $M,g_0$ are considered in more detail.