Definition 3. 2. We conclude that a generalized Riemannian space is an equidistant space if and only if its adjoint Riemannian space is equidistant space. The condition (3. 2) means that the covariant derivative of i (denoted by (; ), (2. 7)) is proportional to the symmetric part of the metric tensor of the space.Jan 31, 2006 Abstract. For this purpose, we introduce the generalized multidimensional scaling, a computationally efficient continuous optimization algorithm for finding the least distortion embedding of one surface into another. The generalized multidimensional scaling algorithm allows for both full and partial surface matching. multidimensional generalized riemannian spaces
A generalized Riemannian space GR N in the sense of Eisenharts denition [5 is a di erentiable Ndimensional manifold, equipped with nonsymmetric basic tensor 1 ij. Let us consider two Ndimensional generalized Riemannian spaces GR N and GR N with basic tensors 1 ij and 1 ij, respectively.
Vol. 90 (2008) Curvature of FourDimensional Generalized Symmetric Spaces 31 The aim of this paper is to describe the curvature of a class of fourdimensional pseudoRiemannian homogeneous spaces, namely, generalized symmetric spaces. As their name indicates, these spaces represent a natural generalization of symmetric spaces. mappings between Riemannian and generalized Riemannian spaces as well as about their invariants (see An Ndimensional manifold MNendowed with a metric tensor Gijnonsymmetric by indices iand j, is the generalized Riemannian space GRN [8. The symmetric and antisymmetric partsmultidimensional generalized riemannian spaces Abstract. Riemannian symmetric spaces have the following two classes of spaces as their natural generalizations: (A) the class ofGSspaces (generalized symmetric Riemannian spaces); (B) the class ofGPSspaces (generalized pointwise symmetric Riemannian spaces). A result due to O. Kowalski says that the relation between the two classes is (A) (B), the inclusion being strict.
Euclidean space. In this case, the Euclidean space is then modeled by the real coordinate space ( Rn) of the same dimension. In one dimension, this is the real line; in two dimensions, it is the Cartesian plane; and in higher dimensions it is a coordinate space with three or more real number coordinates. multidimensional generalized riemannian spaces A Riemannian space is usually understood as a space such that in small domains of it Euclidean geometry holds approximately, to within infinitesimals of higher order in comparison with the dimensions of In particular, a generalized symmetric space is a pseudoRiemannian symmetric space if and only if it admits a regular sstructure of order 2. The order of a generalized symmetric space is the minimum of orders of all possible sstructures on it. Finsler space, generalized. A space with an internal metric, subject to certain restrictions on the behaviour of shortest curves (that is, curves with length equal to the distance between their ends). Such spaces include spaces (see Geodesic geometry) and, in particular, Finsler spaces (cf. II. Multidimensional Generalized Riemannian Spaces geodesic triangles in the following way: K, AP) lim b(T)u(T), 1 TP 169 (0. 1) where b(T) is the excess of the triangle T, that is, the number equal to the sum of its angles minus n, u(T) is the area of the triangle T, that is, the number equal to the area of a Euclidean triangle with the same lengths of sides as T, and