The Reciprocal of the Fundamental Theorem of Riemannian Geometry

2009/05/23 by Héctor H. Calderón, Calderón, Héctor H.
Engineering · Physics and Astronomy · #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Inertial Sensor and Navigation #Relativity and Gravitational Theory #gr-qc

paper · pdf · doi:10.48550/arxiv.0905.3833

6 pages

arxiv created 2009/05/23 · openalex publication_date 2009/05/23 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The fundamental theorem of Riemannian geometry is inverted for analytic Christoffel symbols. The inversion formula, henceforth dubbed Ricardo's formula, is obtained without ancillary assumptions. Even though Ricardo's formula can mathematically give the full answer, it is argued that the solution should be taken only up to a constant conformal factor. A procedure to obtain the Christoffel symbols out of unparameterized geodesics is sketched. Thus, a complete framework to obtain the metric out of measurements is presented. The framework is suitable for analysis of experiments testing the geometrical nature of gravity.

Related

• To the issue of Geodesics and Torsion in Riemannian geometry and Theory of the Gravitation: analysis of consistence and update of the conventional concept
• Riemannian geometry without the hypotheses of homogeneity and symmetry
• The Christoffel problem and two analogs of the Minkowski problem in Riemannian space
• Sectional curvature bounds in gravity: regularisation of the Schwarzschild singularity
• A classical approach to smooth supermanifolds