  In order to facilitate the comparison of Riemannian homogeneous spaces of
compact Lie groups with noncommutative geometries ("quantizations") that
approximate them, we develop here the basic facts concerning equivariant vector
bundles and Dirac operators over them in a way that uses only global
constructions and arguments. Our approach is quite algebraic, using primarily
the modules of cross-sections of vector bundles. We carry the development
through the construction of Hodge--Dirac operators. The inducing construction
is ubiquitous.
