A functional expression for the curvature of hyper-dimensional Riemannian spaces

Abstract

Analogously to the concept of a curvature of curve and surface, in the differential geometry, in the main part of this paper the concept of the curvature of the hyper-dimensional vector spaces of Riemannian metric is generally defined. The defined concept of the curvature of Riemannian spaces of higher dimensions M: M>1, in the further text of the paper, is functional related to the fundamental parameters of an internal geometry of space, more exactly, to components of Riemann-Christoffel's tensor of curvature. At the end, analogously to the concept of lines of curvature in the differential geometry, the concept of sub-spaces of curvature of Riemannian hyper-dimensional vector spaces is also generally defined.