  The theory of spaces with different (not only by sign) contravariant and
covariant affine connections and metrics [}$(\bar{L}_n,g)$\QTR{it}{-spaces] is
worked out within the framework of the tensor analysis over differentiable
manifolds and in a volume necessary for the further considerations of the
kinematics of vector fields and the Lagrangian theory of tensor fields
over}$(\bar{L}_n,g)$\QTR{it}{-spaces. The possibility of introducing affine
connections (whose components differ not only by sign) for contravariant and
covariant tensor fields over differentiable manifolds with finite dimensions is
discussed. The action of the deviation operator, having an important role for
deviation equations in gravitational physics, is considered for the case of
contravariant and covariant vector fields over differentiable manifolds with
different affine connections A deviation identity for contravariant vector
fields is obtained. The notions covariant, contravariant, covariant projective,
and contravariant projective metrics are introduced in
(}$\bar{L}_n,g$\{)-spaces. The action of the covariant and the Lie differential
operators on the different type of metrics is found. The notions of symmetric
covariant and contravariant (Riemannian) connections are determined and
presented by means of the covariant and contravariant metrics and the
corresponding torsion tensors. The different types of relative tensor fields
(tensor densities) as well as the invariant differential operators acting on
them are considered. The invariant volume element and its properties under the
action of different differential operators are investigated.
