Intriguing & intuitive relation of diagonal Riemann tensor components to the corresponding 2-d Gaussian curvature for diagonal metrics of any dimensionality

TL;DR

This paper derives a new, intuitive formula linking diagonal Riemann tensor components to Gaussian curvature in diagonal metrics across any dimension, supported by examples and raising questions about invariant properties.

Contribution

It introduces a novel formula connecting diagonal Riemann tensor components with Gaussian curvature, enhancing understanding of geometric relations in diagonal metrics.

Findings

Derived a formula relating Riemann tensor components to Gaussian curvature

Presented multiple examples demonstrating the formula's application

Raised a question about invariant characteristics of diagonal metrics

Abstract

Using Gauss's square-roots of the metric components, the diagonal Riemann tensor components for diagonal metrics are calculated. The result is a form which makes their source in the metric directly intuitive and displays an intriguing relation to Gaussian curvature. Several examples of calculation utilizing this formula are presented, and a speculative quesiton is raised about the possibility of an invariant characteristic to spaces amenable to orthogonal coordinates/diagonal metrics.