The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi mathvariant="script">C</mml:mi></mml:math>-connection and the 4-dimensional Einstein spaces

We introduce a new family of operators in 4-dimensional pseudo-Riemannian manifolds with a non-vanishing Weyl scalar (non-degenerate spaces) that keep the conformal covariance of conformally covariant tensor concomitants . A particular case that arises naturally is the C -connection that is a Weyl connection that keeps conformal invariance . Using the C − connection we give a new characterization of non-degenerate spaces that are conformal to an Einstein space.