On almost complex Lie algebroids

TL;DR

This paper introduces the concept of almost complex Lie algebroids, explores their properties, and establishes a Newlander-Nirenberg type theorem, along with studying related Hermitian structures and curvature expressions.

Contribution

It defines almost complex Lie algebroids, proves a Newlander-Nirenberg type theorem, and analyzes Hermitian structures and curvature forms on these algebroids.

Findings

Established a Newlander-Nirenberg type theorem for almost complex Lie algebroids

Derived expressions for the $E$-Chern form in terms of the almost complex structure and curvature

Proved conditions under which the Lie algebroid is Hermitian based on curvature and second fundamental form

Abstract

The almost complex Lie algebroids over smooth manifolds are introduced in the paper. In the first part we give some examples and we obtain a Newlander-Nirenberg type theorem on almost complex Lie algebroids. Next the almost Hermitian Lie algebroids and some related structures on the associated complex Lie algebroid are studied. For instance, we obtain that the $E$-Chern form of $E^{1,0}$ associated to an almost complex connection $\nabla$ on $E$ can be expressed in terms of the matrix $J_ER$, where $J_E$ is the almost complex structure of $E$ and $R$ is the curvature of $\nabla$. Also, we consider a metric product connection associated to an almost Hermitian Lie algebroid and we prove that the mean curvature section of $E^{0,1}$ vanishes and the second fundamental $2$--form section of $E^{0,1}$ vanishes iff the Lie algebroid is Hermitian.