Systems of symplectic forms on four-manifolds

TL;DR

This paper classifies certain four-dimensional manifolds with special symplectic structures, focusing on almost Hermitian and almost Kähler manifolds with restricted holonomy, providing a comprehensive understanding of their geometric properties.

Contribution

It offers a classification of Riemannian 4-manifolds with five orthonormal symplectic forms and fully describes almost Kähler 4-manifolds within this framework.

Findings

Classification of 4-manifolds with symplectic forms

Description of almost Kähler 4-manifolds

Holonomy algebra constraints on structures

Abstract

We study almost Hermitian 4-manifolds with holonomy algebra, for the canonical Hermitian connection, of dimension at most one. We show how Riemannian 4-manifolds admitting five orthonormal symplectic forms fit therein and classify them. In this set-up we also fully describe almost Kaehler 4-manifolds.