Clifford and extensor calculus and the riemann and ricci extensor fields of deformed structures (M,del ',eta) and (M,del,g)

摘要

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 a triple ( M,del, g) is investigated for each particular open set U subset 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 to 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.