Let M be a compact

differentiable manifold with universal covering p : [~.M] [right arrow] M and a finite covering [bar.p] : [~.M] [right arrow] M with the natural map p' : [~.M] [right arrow] [bar.M] with p = p' [??] [bar.p].

In [9] Friedmann and Schouten introduced the notion of semi-symmetric linear connection on a

differentiable manifold. Then in 1932 Hayden [12] introduced the idea of metric connection with torsion on a Riemannian manifold.

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

According to the concepts of topological manifold and

differentiable manifold and based on the discontinuous dynamics of DDA block systems, the numerical manifold method (NMM) adopts the finite cover technique to set up a unified calculation form for the finite element method, the DDA, and the analytical method [1].

GL(n) is a Lie group, that is, a group which is also a

differentiable manifold and for which the operations of group multiplication and inverse are smooth.

Let [??] be an almost contact metric manifold of dimension 2[??] + 1, that is, a (2[??] + 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. An almost complex structure on M is a tensor field J of type (1, 1) on M such that [J.sup.2] = -I.

The standard concept of a '

differentiable manifold' as to be found in e.g.

(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.) Thus, we can resist the temptation to suppose that substantivalism is implicit in Newton's treatment of space as a continuum.(2)

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.

Let [M.sup.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.sub.p]: [T.sub.p]M x [T.sub.p]M [right arrow] R is an inner product of signature (-, +, +, ..., +), where [T.sub.p]M denotes the tangent vector space of M at p and R is the real number space which satisfies