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

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.