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 , 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
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  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