This paper presents a thoughful review of: (a) the Clifford algebra (Formula presented.) of multivecfors which is naturally associated with a hyperbolic space HV ; (b) the study of the properties of the duality product of multivectors and multiforms; (c) the theory of k multivector and l multiform variables multivector extensors over V and (d) the use of the above mentioned structures to present a theory of the parallelism structure on an arbitrary smooth manifold introducing the concepts of covariant derivarives, deformed covariant derivatives and relative covariant derivatives of multivector, multiform fields and extensors fields. © 2014 Springer Basel.