TENSORIAL CURVATURE AND D-DIFFERENTIATION PART I: "COMMUTATIVE" KIND

Abstract

A special class of operators of D-differentiation is introduced, called the "commutative" kind. It is closely related to the family of D-differentiation operators, the curvature of which is a tensor (as opposed to a non-linear operator), and to that of the D-differentiation operators admitting a scalar curvature. It is found that all commutative D-differentiation operators admit a scalar curvature, but that only a proper subset of them (which is explicitly characterized) has a curvature operator that is a tensor. It is also established that all commutative D-differentiation operators can be expressed in terms of covariant differentiation and a tensor field. This generalizes the well-known result about the difference of two sets of connection coefficients yielding a tensor field. In a companion article, a cognate class of operators is defined, which contains the commutative type as a special case. It enables one to construct a unified framework for the Einstein–Maxwell theory through D-differentiation.