  Here (the last paper in a series of four) we end our presentation of the
basics of a systematical approach to the differential geometry of a smooth
manifold M (supporting a metric field g and a general connection del) which
uses the geometric algebras of multivector and extensors (fields) developed in
previous papers. The theory of the Riemann and Ricci fields of the the triple
(M,del,g)is investigated to for each particular open set U of M through the
introduction of a geometric structure on U, i.e., a triple (U,gamma,g) where
gamma is a general connection field on U and g is a metric extensor field
associated to g. The relation between geometrical structures related by gauge
extensor fields is clarified. These geometries may be said to be deformations
one of each other. Moreover we study the important case of a class of deformed
Levi-Civita geometrical structures and prove key theorems about them that are
important in the formulation of geometric theories of the gravitational field.
