Dual left-invariant almost complex structures on the $SU(2)\times SU(2)$

TL;DR

This paper investigates the properties of dual left-invariant almost complex structures on the manifold SU(2)×SU(2), focusing on conditions under which certain 3-forms define almost complex structures and analyzing their characteristics.

Contribution

It derives conditions for an almost complex structure to be defined by a non-degenerate 3-form on SU(2)×SU(2) and explores the properties of these structures.

Findings

Conditions for $J_I$ to be an almost complex structure are established.

Properties of the dual structures $J_I$ are characterized.

The relationship between 3-forms and almost complex structures on SU(2)×SU(2) is clarified.

Abstract

There exist non-degenerate 3-form $d\omega_I$, $\omega_I(X,Y)=g(IX,Y)$, for each leftinvariant almost Hermitian structure $(g,I)$, where $g$ is Killing-Cartan metric on the $M=S^3\times S^3=SU(2)\times SU(2)$. Known \cite{H1}, that arbitrary non-degenerate 3-form on the 6-dimensional manifold, with some additional properties defines the almost complex structure. Condition for $I$ to define almost complex structure $J_I$ by $d\omega_I$ is obtained. Properties of $J_I$ are researched.