The essential regularity of singular connections in geometry

2024/12/12 by Moritz Reintjes, Blake Temple, Reintjes, Moritz +1 · 1 citation
Engineering · Mathematics · #58K30 #83C75 #Advanced Numerical Analysis Techniques #Advanced Theoretical and Applied Studies in Material Sciences and Geometry #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Mathematical Physics (math-ph) #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.2412.08928

openalex publication_date 2024/12/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper, a culmination of the authors' theory of the RT-equations, accomplishes the following: (i) We discover there is a true (geometric) regularity associated with every affine connection, its ``essential regularity'', the highest possible regularity achievable by coordinate transformation, a geometric property independent of starting atlas. (ii) We give a checkable necessary and sufficient condition for determining whether or not a connection is at its essential regularity in a given atlas, based on the relative regularity of the connection and its Riemann curvature. (iii) We introduce a computable procedure based on the RT-equations for lifting any L^p affine connection given in a starting atlas, to a new atlas in which the connection exhibits its essential regularity. This resolves the long-standing problem of determining whether or not a singularity in an affine connection is removable or essential, applicable to any connection with components locally in L^p, p>n, general enough to include GR shock wave and cusp singularities in General Relativity. Since a manifold by itself does not carry an intrinsic level of regularity, the authors propose that the essential regularity of a connection marks the point at which an intrinsic level of regularity enters the subject of geometry.

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