Metric tensor definition model for service-oriented network based on curvilinear coordinates systems

In this paper we analyzed definitions of the metric tensor, Christopher's symbols, Riemann and Ricci tensors, scalar curvature of space for different spaces. The component of the metric tensor is determined by using the cosine theorem for a quadrilateral, taking into account the two-way connection between each pair of nodes. We proposed the way to increase the number of components of a metric tensor that will allow to represent a metric in a symmetric tensor space and describe the deformation of the Riemann metric which is used in the Ricci flows.