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The discourse begins with a definition of a Lie algebroid which is a vector bundle $p : A \to M$ over a manifold with an $R$-Lie algebra structure on the smooth section module and a bundle morphism $\gamma : A \to TM$ which induces a Lie algebra morphism on the smooth section modules. If $\gamma$ has constant rank, the Lie algebroid is called regular. (A monograph on the theory of Lie groupoids and Lie algebroids is published by {\it K. Mackenzie} [Lie groupoids and Lie algebroids in differential geometry (1987; Zbl 0683.53029)].) A principal $G$-bundle $(P, \pi, M, G, \cdot)$ gives rise to Lie algebroid $A(P)$. Since every vector bundle determines a $\text {GI} (n)$-principal bundle, it also determines a Lie algebroid. Many other examples illustrate the fact that Lie algebroids are a prevalent phenomenon. The author's survey describes a theory of connections for regular Lie algebroids over a manifold equipped with a constant dimensional smooth distribution, and a! (English) |
