Speaker
Description
Lie algebroids provide a unifying framework that extends several classical
structures of differential geometry, including tangent bundles, Lie algebras
of vector fields, and foliations. The aim of this contribution is to present
Lie algebroids from an introductory and conceptually transparent perspective,
emphasizing how they arise naturally from familiar geometric objects.
We begin by reviewing the necessary background material, such as vector
bundles, sections, tangent and cotangent bundles, bundle maps, and pullbacks.
We also recall basic operations on differential forms, including the exterior
derivative, interior product, and Lie derivative, which play a fundamental
role in the formulation of Lie algebroid structures.
After introducing the definition of a Lie algebroid, we discuss its two main
components: the anchor map and the Lie bracket on sections. We explain how
these structures generalize the classical Lie bracket of vector fields and
clarify the geometric meaning of the anchor map. Different types of Lie
algebroids are briefly discussed through the properties of the anchor.
As a concrete example, we construct a Lie algebroid structure on the
derivation bundle of a vector bundle. This example illustrates how Lie
algebroids naturally emerge from standard differential-geometric constructions
and highlights their role as a natural extension of classical geometry.
Affiliation
Department of Mathematics, Gebze Technical University, Kocaeli, Türkiye