Spaces with contravariant and covariant affine connections and metrics

The theory of spaces with contravariant and covariant affine connections, whose components differ not only in sign, and metrics [(Ln,g) spaces] is worked out within the framework of tensor analysis over differentiable manifolds and in a volume necessary for further consideration of the kinematics of vector fields and the Lagrangian theory of tensor fields over (Ln ,g) spaces. The possibility of introducing affine connections for contravariant and covariant tensor fields, whose components differ not only in sign, 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 (called Ln spaces). A deviation identity for contravariant vector fields is obtained. The notions of covariant, contravariant, covariant projective, and contravariant projective metric are introduced in (Ln,g) spaces. The action of the covariant and the Lie differential operator on the different types of metric is found. The concepts of symmetric covariant and contravariant (Riemannian) connection are defined and presented by means of the covariant and contravariant metric 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.