Riemann and Ricci Fields in Geometric Structures

Here (the last paper in a series of eight) we end our presentation of the basics of a systematical approach to the differential geometry of smooth manifolds which uses the geometric algebras of multivector and extensors (fields) developed in previous papers. The theory of the Riemann and Ricci fields associated to a given geometric structure, i.e., a triple (M,γ, g) where M is a smooth manifold, γ is a general connection field and g is a metric extensor field is scrutinized. 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.