On almost complex Lie algebroids
Cristian Ida, Paul Popescu

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 -Chern form of associated to an almost complex connection on can be expressed in terms of the matrix , where is the almost complex structure of and is the curvature of . Also, we consider a metric product connection associated to an almost Hermitian Lie algebroid and we prove that the mean curvature section of vanishes and the second fundamental --form section of vanishes iff the Lie algebroid is Hermitian.
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Taxonomy
TopicsAdvanced Topics in Algebra · Homotopy and Cohomology in Algebraic Topology · Nonlinear Waves and Solitons
