An n-dimensional

differentiable manifold M is said to admit an almost para-contact Riemannian structure ([phi], [xi], [eta], g), where [phi] is a (1, 1) tensor field, [xi] is a vector field, [eta] is a 1-form and g is a Riemannian metric on M such that

n] be an n-dimensional

differentiable manifold endowed with a (1,1) tensor field [phi], a contravariant vector field [xi], a covariant vector field [eta] and a Lorentzian metric g of type (0, 2) such that for each point p [member of] M, the tensor [g.

1)-dimensional

differentiable manifold endowed with an almost contact metric structure ([phi], [xi], [eta] g).

Let (M, L) be a Finsler space, where M is an n-dimensional

differentiable manifold associated with the fundamental function L.

Let M be a real 2m-dimensional

differentiable manifold.

Next, let M be a real (2n + 1)-dimensional

differentiable manifold, endowed with an almost contact metric structure (f, [xi], [eta], g).

As we have suggested, a rate distortion manifold need not necessarily be a

differentiable manifold in the conventional sense, but may admit an abstract differentiable space structure (such as that described in Appendix III).

A trivial example of a

differentiable manifold is any open subset of a Euclidean space, so both [P.

This is analogous to the situation in modern spacetime theory, in which we introduce an infinitely

differentiable manifold so that the derivatives named in dynamical equations can be assumed to exist.

An alternative to the usual approach via the Frobenius integrability conditions was proposed in an article of 1972 in which I defined a differentiable preference relation by the requirement that the indifferent pairs of commodity vectors from a

differentiable manifold.

A non flat n-dimensional

differentiable manifold [M.

A semi-symmetric linear connection in a

differentiable manifold was introduced by Friedmann and Schouten in [5].