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

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.