On Almost Complex Lie Algebroids

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 ∇ on E can be expressed in terms of the matrix JER, where JE is the almost complex structure of E and R 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 E 0,1 vanishes and the second fundamental 2-form section of E 0,1 vanishes iff the Lie algebroid is Hermitian.

Laplace operators on holomorphic Lie algebroids

“…We will define two Laplace operators for functions, a horizontal and a vertical one. The horizontal Laplace operator for functions on the prolongation algebroid is 6) and the vertical one is…”

“…Real Lie algebroids have been studied by A. Weinstein [21], P. Popescu [17,18], M. Anastasiei [2], L. Popescu [20]. Complex and holomorphic Lie algebroids have been investigated by C.-M. Marle [11], P. Popescu [19], P. Popescu, C. Ida [6]. E. Martinez [12,13] has introduced the notion of prolongation of a Lie algebroid, as a tool for studying the geometry of a Lie algebroid in a context which is similar to the tangent bundle of a manifold.…”

Almost Lie Algebroids and Characteristic Classes

“…Analogously to the Lie algebroid case [11,19], an almost complex almost Lie algebroid is a real almost Lie algebroid E such that there is an almost complex endomorphism on E (i.e., an endomorphism…”

Finsler structures on holomorphic Lie algebroids

“…The following definition is known from [6,7,10,12,20]. A holomorphic Lie algebroid over M is a triple (E, [·, ·] E , ρ E ), where E is a holomorphic vector bundle anchored over M , [·, ·] E is a Lie bracket on Γ(E) and ρ E : Γ(E) → Γ(T M ) is the homomorphism of complex modules induced by the anchor map ρ such that…”