Clifford and Extensor Calculus and the Riemann and Ricci Extensor Fields of Deformed Structures

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 ∇) 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,∇,g) is investigated to for each particular open set U ⊂ M through the introduction of a geometric structure on U , i.e., a triple (U, γ, g) where γ 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.