Conformal Form of Pseudo-Riemannian Metrics by Normal Coordinate Transformations

2010/06/15 by A. C. V. V. de Siqueira, de Siqueira, A. C. V. V.
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Mathematical Physics (math-ph) #Relativity and Gravitational Theory #math-ph #math.MP

paper · pdf · doi:10.48550/arxiv.1006.2868

27 pages, no figures

arxiv created 2010/06/15 · openalex publication_date 2010/06/15 · arxiv updated 2010/06/16 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

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.

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