Conformal Form of Pseudo-Riemannian Metrics by Normal Coordinate Transformations II

TL;DR

This paper revisits conformal geometry for pseudo-Riemannian metrics, demonstrating their conformality to flat or constant curvature manifolds via normal coordinate transformations, and explores implications for classical and quantum angular momenta.

Contribution

It introduces a novel approach to conformal geometry based on Cartan's method, showing all pseudo-Riemannian metrics are conformal to flat or constant curvature spaces and can be embedded in higher-dimensional cones.

Findings

All pseudo-Riemannian metrics are conformal to flat or constant curvature manifolds.

Metrics can be embedded in a hyper-cone of a flat n+2-dimensional space.

New interpretations of classical and quantum angular momenta are provided.

Abstract

In this paper, we have reintroduced a new approach to conformal geometry developed and presented in two previous papers, in which we show that all n-dimensional pseudo-Riemannian metrics are conformal to a flat n-dimensional manifold as well as an n-dimensional manifold of constant curvature when Riemannian normal coordinates are well-behaved in the origin and in their neighborhood. This was based on an approach developed by French mathematician Elie Cartan. As a consequence of geometry, we have reintroduced the classical and quantum angular momenta of a particle and present new interpretations. We also show that all n-dimensional pseudo-Riemannian metrics can be embedded in a hyper-cone of a flat n+2-dimensional manifold.