Spaces with contravariant and covariant affine connections and metrics

Abstract

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

Flows and particles with shear-free and expansion-free velocities in (bar-Lng)- and Weyl spaces

“…(L n , g), Y n , U n , and V n are spaces with contravariant and covariant affine connections and metrics whose components differ not only by sign [7]. In such type of spaces the non-canonical contraction operator S acts on a contravariant basic vector field e j (or ∂ j ) ∈ {e j (or ∂ j )} ⊂ T (M ) and on a covariant basic vector field…”

“…On the other side, the results of the action of the covariant differential operator ∇ u and of the Lie differential operator £ u on the invariant volume element dω are recurrent relations for dω [7] …”

“…£ u is the Lie differential operator [7] acting on the elements of the tensor algebra T over M . The action of £ u is called dragging-along a contravariant vector field u.…”

“…The use of spaces with affine connections and metrics as models of space-time has been critically evaluated from different points of view [5], [6]. But recently, it has been proved that in spaces with contravariant and covariant affine connections (whose components differ only by sign) and metrics [(L n , g)-spaces] as well as in spaces with contravariant and covariant affine connections (whose components differ not only by sign) and metrics [(L n , g)-spaces] [7] [8]…”

On the Existence of a Gyroscope in Spaces with Affine Connections and Metrics

“…Let a contravariant affine connection Γ and a covariant affine connection P be given [3] with components in a co-ordinate basis given respectively as Γ i jk and P i jk over a differentiable…”