The essential regularity of singular connections in geometry

Abstract This paper, a culmination of the authors’ theory of the Regularity Transformation (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 Lp 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 Lp, p>n, general enough to include singularities at shock waves 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.