Laplace operators on holomorphic Lie algebroids

Abstract

The paper introduces Laplace-type operators for functions defined on the tangent space of a Finsler Lie algebroid, using a volume form on the prolongation of the algebroid. It also presents the construction of a horizontal Laplace operator for forms defined on the prolongation of the algebroid. All of the Laplace operators considered in the paper are also locally expressed using the Chern-Finsler connection of the algebroid.

The warped product of holomorphic Lie algebroids

“…Let us first recall the construction of the holomorphic prolongation T E. For more details, see [16,17,18]. Let E be a holomorphic Lie algebroid over a complex manifold M .…”

“…In [18], we have introduced the gradient of a function on the holomorphic prolongation T E. Now, we are interested in the gradient defined on the bundle TE, i.e., the operator ∇ given by…”

“…Since the prolongation bundle has similar properties to the tangent bundle of a manifold, as we have discussed in some previous papers [16,17,18], we will further work in this setting instead of the tangent bundles of E 1 and E 2 . We easily find that T(E 1 × E 2 ) = TE 1 × TE 2 .…”

“…Following the line of the study of holomorphic Lie algebroids and their prolongations, initiated in [16,17,18,19], in the present paper we investigate the notion of warped product E 1 × f E 2 of two algebroids E 1 and E 2 .…”

Finsler structures on holomorphic Lie algebroids

“…We briefly recall here some general notions and set our notations for the geometry of holomorphic Lie algebroids. More ideas can be found in [8,9].…”