(Wydawnictwo Politechniki Łódzkiej, 2021) Balcerzak, Bogdan; Wolak, Robert A.; Sasin, Wiesław
Lie algebroids appear in many structures related to geometry. Although
the motivations for defining the concept of Lie algebroid come from Lie
groupoids [74], on the one hand, we can view them as some generalizations of a tangent bundle or integrable distribution on a differential
manifold, and on the other hand, as a generalization of Lie algebra.
Further, the structures of the Lie algebroid can be generalized to, for example, structures in which the Lie bracket does not satisfy the Jacobi condition or at all the structure without the Lie bracket, however,
equipped with a morphism acting from a given vector bundle into a tangent bundle (called an anchor). A vector bundle equipped with an
anchor allows us to introduce the concept of connection. Our considerations focus on linear connections and their properties, and on the
existence of a connection in a given vector bundle compatible with an existing metric structure.[...]
