  In this paper, we study Absolute Parallelism (AP-) geometry on the tangent
bundle $TM$ of a manifold $M$. Accordingly, all geometric objects defined in
this geometry are not only functions of the positional argument $x$, but also
depend on the directional argument $y$. Moreover, many new geometric objects,
which have no counterpart in the classical AP-geometry, emerge in this
different framework. We refer to such a geometry as an Extended Absolute
Parallelism (EAP-) geometry. The building blocks of the EAP-geometry are a
nonlinear connection assumed given a priori and $2n$ linearly independent
vector fields (of special form) defined globally on $TM$ defining the
parallelization. Four different $d$-connections are used to explore the
properties of this geometry. Simple and compact formulae for the curvature
tensors and the W-tensors of the four defined $d$-connections are obtained,
expressed in terms of the torsion and the contortion tensors of the EAP-space.
Further conditions are imposed on the canonical $d$-connection assuming that it
is of Cartan type (resp. Berwald type). Important consequences of these
assumptions are investigated. Finally, a special form of the canonical
$d$-connection is studied under which the classical AP-geometry is recovered
naturally from the EAP-geometry. Physical aspects of some of the geometric
objects investigated are pointed out and possible physical implications of the
EAP-space are discussed, including an outline of a generalized field theory on
the tangent bundle $TM$ of $M$
