The essential regularity of singular connections in Geometry

TL;DR

This paper introduces a geometric notion of essential regularity for affine connections, providing a checkable criterion and a computable procedure to lift connections to this regularity, with applications to singularities in geometry and physics.

Contribution

It establishes the concept of essential regularity for affine connections, offers a criterion to identify it, and develops a procedure to achieve it from less regular connections, advancing the understanding of singularities in geometry.

Findings

Discovered the geometric regularity of affine connections independent of coordinate charts.

Provided a necessary and sufficient condition based on curvature for essential regularity.

Developed a computable iterative procedure to lift connections to their essential regularity.

Abstract

We accomplish three things: (i) We discover the geometric (true) regularity of affine connections, their essential (highest possible) regularity, a geometric property independent of starting atlas. (ii) We give a checkable necessary and sufficient condition for determining whether or not connections are at their essential regularity, based on the relative regularity of the connection and its Riemann curvature. (iii) We introduce a computable procedure for lifting any $L^p$ affine connection in an atlas ($p>n$), to a new atlas in which the connection exhibits its essential regularity. To accomplish this, we prove that the RT-equations, originally designed by the authors to locally lift the regularity of singular connections by one derivative, surprisingly, also induce an implicit hidden regularization of the Riemann curvature, together with a global regularization of transition maps between regularizing coordinate charts. From this we deduce a multi-step regularization of the connection, and construct a new atlas in which the connection exhibits its essential regularity. This paper is a culmination of the theory of the RT-equations which provides a computable iterative procedure for lifting an atlas to a new atlas in which the connection exhibits its essential regularity, applicable to any $L^p$ affine connection defined in a $W^{2,p}$ starting atlas, $p>n$. This provides a definitive theory for determining whether singularities in an $L^p$ affine connection are essential or removable by coordinate transformation, together with an explicit procedure for lifting removable singularities to their essential regularity, both locally and globally, $p>n$. This includes GR shock wave singularities and cusp singularities (continuous metrics with infinite gradients) in General Relativity. The essential regularity is the point where an intrinsic level regularity enters the subject of Geometry.