Finsler structures on holomorphic Lie algebroids

Abstract

Complex Finsler vector bundles have been studied mainly by T. Aikou, who defined complex Finsler structures on holomorphic vector bundles. In this paper, we consider the more general case of a holomorphic Lie algebroid and we introduce Finsler structures, partial and Chern-Finsler connections on it.First, we recall some basic notions on holomorphic Lie algebroids. Then, using an idea from E. Martinez, we introduce the concept of complexified prolongation of such an algebroid. Also, we study nonlinear and linear connections on the tangent bundle T C E and on the prolongation T C E and we investigate the relation between their coefficients. The analogue of the classical Chern-Finsler connection is defined and studied in the paper for the case of the holomorphic Lie algebroid E.

Laplace operators on holomorphic Lie algebroids

“…In [8], we have considered an adapted frame on T E given by a complex nonlinear connection. In [9], we have introduced a complex nonlinear connection of Chern-Finsler type on the holomorphic prolongation T E. Here we only recall the notions we need for defining Laplace type operators on the holomorphic Lie algebroid.…”

“…In [9], following the ideas from [1], we have introduced the Chern-Finsler nonlinear connection of the prolongation T E. If F : E → R + is a Finsler function on E ([9]), i.e. it is homogeneous, and the complex Finsler metric tensor h αβ =∂ α∂β F, is strictly pseudoconvex, then N β α = hσ β ∂ α∂σ F are the coefficients of the Chern-Finsler nonlinear connection of the prolongation T E. Also, a Chern-Finsler linear connection of type (1, 0) on T E is given by…”

“…see also [9] for more details. If we denote by h = det(h αβ ), then using similar reasons as in the case of a complex Finsler manifold ( [23]) we get…”

“…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. In this paper, we use this setting in complex geometry to continue the study of holomorphic Lie algebroids from [7,8,9] for introducing Laplace-type operators for functions and for forms on a Finsler algebroid.…”

“…The first section briefly recalls notions from the geometry of Finsler Lie algebroids ( [7,8]) which will be used in defining the Laplace operators. The second section presents the case of Finsler algebroids ( [9]), when a Chern-Finsler connection is defined from a Finsler function on the algebroid. The third section introduces two Laplacians for functions, a horizontal and a vertical one, following the ideas from the case of a complex Finsler manifold ( [23]).…”