The Main Theorem for (Anti-)Self-Dual Conformal Torsion-Free Connection on a Four-Dimensional Manifold

Abstract

In this paper we obtain results that occur on a four-manifold with conformai torsion-free connection for all possible signatures of angular metric. It is proved that three of the four terms of the formula for the decomposition of the basic tensor are equidual, one is anti-dual. Based on this result we find conditions for (anti-)self-duality of exterior 2-forms Φ i j , i, j = 1, 2, 3, 4, which are part of components of the conformal curvature matrix. With the help of the latter result, the main theorem is proved: a conformal torsion-free connection on a four-manifold with the signatures of the angular metric s = ±4; 0 is (anti-)self-dual if and only if the Weyl tensor of the angular metric and the exterior 2-form Φ 0 0 are (anti-)self-dual and Einstein’s and Maxwell’s equations are satisfied. In particular, the normal conformal Cartan connection is (anti-)self-dual if and only if such is also the Weyl tensor of the angular metric.