The theory of spaces with different (not only by sign) contravariant and covariant affine connections and metrics [ (Ln, g)

The notions of the Fermi covariant differential operator with its corresponding Fermi derivative and Fermi-Walker transport are generalized for the case of differentiable manifolds with an affine connection and a metric, (L n , g)-spaces. The generalization of Fermi-Walker transport is not unique and depends on the structure of the covariant antisymmetric tensor of second rank in the construction of the Fermi-Walker transport. The existence of such a type of transport over (L n , g)-spaces allows the determination of a proper non-rotating accelerated observer's frame of reference, if an (L n , g)-space is used as a model of the spacetime.

The notions of Fermi covariant differential operator, Fermi derivative, and Fermi–Walker transport are generalized for the case of differentiable manifolds with different (not only by sign) contravariant and covariant affine connections and metrics [[Formula: see text]-spaces]. The generalization of Fermi–Walker transport is not unique and depends on the structure of the covariant antisymmetric tensor of second rank in the construction of the Fermi–Walker transport. The existence of such type of transport over [Formula: see text]-spaces allows the determination of a proper nonrotating accelerated observer's frame of reference, if a [Formula: see text]-space is used as a model of the space–time.

The notions of ordinary, covariant and Lie differentials are considered as operators over differentiable manifolds with different (not only by sign) contravariant and covariant affine connections and metric. The difference between the interpretations of the ordinary differential as a covariant basic vector field and as a component of a contravariant vector field is discussed. By means of the covariant metric and the ordinary differential the notion of the line element is introduced and the geodesic equation is obtained and compared with the autoparallel equation.