The essential regularity of singular connections in Geometry

In this paper 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 just one derivative, surprisingly, also induce an implicit hidden regularization of the Riemann curvature tensor, 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. The present paper is a culmination of the theory of the RT-equations which provides a definitive theory for determining whether or not singularities in an $L^p$ affine connection ($p>n$) are essential or removable by coordinate transformation, together with an explicit procedure for lifting removable singularities to their essential regularity, both locally and globally. This includes GR shock wave singularities and cusp singularities (continuous metrics with infinite gradients), both obstacles to numerical simulation. We propose the essential regularity of a connection to be the point where an intrinsic level of regularity enters the subject of Geometry.