Conformal Form of Pseudo-Riemannian Metrics by Normal Coordinate Transformations

TL;DR

This paper extends Cartan's approach to show that all pseudo-Riemannian metrics are conformal to flat or constant curvature manifolds, enabling a geometric derivation of classical and quantum angular momenta.

Contribution

It introduces a conformal form for pseudo-Riemannian metrics using normal coordinate transformations, linking geometry to angular momentum without additional postulates.

Findings

All pseudo-Riemannian metrics are conformal to flat manifolds in normal coordinates.

Metrics can be embedded in a hyper-cone of a higher-dimensional flat space.

Classical and quantum angular momenta are derived from geometric principles.

Abstract

In this paper we extend the Cartan's approach of Riemannian normal coordinates and show that all n-dimensional pseudo-Riemannian metrics are conformal to a flat manifold, when, in normal coordinates, they are well-behaved in the origin and in its neighborhood. We show that for this condition all n-dimensioanl pseudo-Riemannian metrics can be embedded in a hyper-cone of an n+2-dimensional flat manifold. Based on the above conditions we show that each n-dimensional pseudo-Riemannian manifolds is conformal to an n-dimensional manifold of constant curvature. As a consequence of geometry, without postulates, we obtain the classical and the quantum angular momenta of a particle.