The Gap metric is proved to be more suitable for measuring the distance between two linear systems than the norm-based ones [7, 8], and the effect of dimension on each variable can be reflected in the

Riemannian space when data preprocessing is performed.

It is well known that, for a

Riemannian space [V.sub.n], the maximal group of motions is of the order less than or equal to n(n+1)/2.

In the same framework he solves the famous problem of the prolongation of order k > 1 of the

Riemannian space, brings a solid contribution to the foundation of the Mechanics of the Lagrangians which depends on the higher order accelerations and creates some new geometrical models for the theory of physical fields.

The dimension of the flat tangent space and the correlation between the imaginary and real basis vectors are the same as in the corresponding

Riemannian space. A system of basis vectors [e.sub.[alpha]] can be introduced at any point of the locally tangent space.

Miyazawa, "On

Riemannian space with recurrent conformal curvature," Tensor, vol.

Otsuki's method is generalized by many authors to study hypersurfaces with constant k-th mean curvature and two distinct principal curvatures in

Riemannian space forms (see e.g., [6], [9], [12]) or spacelike hypersurfaces in de Sitter space (see e.g., [5], [7], [10], [14]).

The first (16) depends only on Christoffel symbol of the second kind, which coincides with Riemannian-Christoffel curvature tensor of the

Riemannian space. The second (17) depends only on the contortion (or torsion via (11)) of the PAP-space.

In an Euclidean geometric space the proposition G is [true.bar]; in a Riemannian geometric space the proposition G is [false.bar] (since there is no parallel passing through an exterior point to a given line); in a Smarandache geometric space (constructed from mixed spaces, for example from a part of Euclidean subspace together with another part of

Riemannian space) the proposition G is [indeterminate.bar] (true and false in the same time).

For instance a continuous and smooth image can be considered as a

Riemannian space (a smooth surface) whose metric depends on the local gradient.

In all these theories M is the configuration space, T is the kinetic energy of a

Riemannian space [R.sup.n] = (M,g), F(x,y) is the fundamental function of a Finsler space [F.sup.n] = (M, F(x,y)), L(x,y) is a regular Lagrangian and Fe(x, y) are the external forces which depend on the material points x [member of] M and their velocities [y.sup.i] = [x.sup.i] = [d[x.sup.i]/dt].

Consider the embedding of a four-dimensional

Riemannian space parametrized by local coordinates (denoted by [x.sup.[mu]], [mu], V = 0,1,2,3) in a pseudo-Euclidean space [E.sup.N] of dimension N.

and Miyazawa, T., On a

Riemannian space with recurrent conformal curvature, Tensor(N.S.), 18 (1967), 348-354.