Geometry of Lie-admissible algebras

The first portion of this paper consists of basic definitions and results on manifolds, fiber bundles, connections and covariant differentiation. The purpose of this is to identify a relationship between the geometries of manifolds and some classes of algebras; in particular the class of Lie-admissible algebras. Some indication is given of how results in the theory of connections can be applied to the study of Lie-admissible algebas. In the second part, some results on reductive Lie-admissible algebras are given that should have implications for the decomposition of manifolds