The article analyzes geometric structures on smooth inner product vector bundles determined by geometric algebras,
a class that includes matrix, group with two–cocycle,
and Euclidean Clifford algebras. Part one of the article introduces, as
requisites, the concepts of algebra environments, structure
manifolds, Zariski tangent spaces, and derivations on algebra
environments. Part two provides the definitions of geometric
algebras and their associated Hom environments.
The main result identifies the structure manifolds of Hom environments with spaces of algebra homomorphisms.
Part three develops an algebraic approach to the study of
geometric structures on smooth vector bundles, and concludes with
characterizations of derivations and linear connections that
preserve prescribed structures.
