  This paper, sixth in a series of eight, uses the geometric calculus on
manifolds developed in previous papers of the series to introduce through the
concept of a metric extensor field g a metric structure for a smooth manifold
M. The associated Christoffel operators, a notable decomposition of that object
and the associated Levi-Civita connection field are given. The paper introduces
also the concept of a geometrical structure for a manifold M as a triple
(M,g,gamma), where gamma is a connection extensor field defining a parallelism
structure for M . Next, the theory of metric compatible covariant derivatives
is given and a relationship between the connection extensor fields and
covariant derivatives of two deformed (metric compatible) geometrical
structures (M,g,gamma) and (M,eta,gamma') is determined.
