Finally, in Section 4, we examine

Riemannian optimization algorithms for the Karcher mean, which is defined as the minimizer over all positive definite matrices of the sum of squared (intrinsic) distances to all matrices in the mean.

Ricci solitons are introduced as triples (M, g, V), where (M, g) is a

Riemannian manifold and V is a vector ield so that the following equation is satisied:

The geometry of slant and semi-slant submanifolds in metallic

Riemannian manifolds is related by the properties of slant and semi-slant submanifolds in almost product

Riemannian manifolds, studied in ([7, 8,16]).

where S is a surface of a 3-dimensional

Riemannian manifold N, H denotes the mean curvature of S, and [PHI] [member of] [C.sup.[infinity]] (N).

and the structure is metric with respect to a generalized

Riemannian metric G if

The features are extracted by using the

Riemannian geometry [14], which manipulates the covariance matrices of the MEG signals.

The paper is organized as follows: In Section 1 we recall the basic definitions and properties of homogeneous geodesics in a

Riemannian manifold.

Metrics of

Riemannian spaces are symmetric ([g.sub.[alpha][beta]] = [g.sub.[beta][alpha]]) and non-degenerate (g = det [parallel][g.sub.[alpha][beta]][parallel] [not equal to] 0), while the elementary four-dimensional interval is invariant relative to any reference system ([ds.sup.2] = const).

The energy of a differentiable map f: (M, g) [right arrow] (N, h) between

Riemannian manifolds is given by

On the existence of an ACV and specific restrictions on the ambient manifold [bar.M], we recall that, in 1923, Eisenhart [10] proved that "If a

Riemannian manifold [bar.M] admits such a tensor K, independent of [bar.g], then [bar.M] is reducible." This means that [bar.M] is locally a product manifold of the form ([bar.M] = [M.sub.1] x [M.sub.2], [bar.g] = [g.sub.1] [cross product] [g.sub.2]) and there exists a local coordinate system in terms of which the distance element of g is given by

In other words, its acceleration is normal to the manifold so that the geodesic curvature is zero along the geodesic, and thus the two-point boundary value problem (TPBVP) arises from geodesic differential equations on

Riemannian manifold.