On the real theory of four-dimensional conformal structures

The conformal structures CO(4,0), CO(1,3) and CO(2,2) are studied on a real manifold M, dim M = 4. On M isotropic fiber bundles E(alpha) and E(beta) are constructed. These bundles are real for the CO(2,2)-structure, and they satisfy the condition <(E)over bar (alpha)> = E(beta) for the CO(1,3)-structure, and the conditions <(E)over bar (alpha)> = E(alpha), <(E)over bar (beta)> = E(beta) for the CO(4)-structure. The tensor C of conformal curvature splits into two subtensors C-alpha and C-beta which are the curvature tensors of the bundles E(alpha) and E(beta), respectively. These subtensors satisfy the same conditions as the bundles E(alpha) and E(beta). Con formally semiflat and flat structures and their geometrical characteristics are studied. The principal 2-directions are defined, and conditions for their integrability are obtained. These investigations for the CO(1,3)-structure are connected with Petrov's classification of Einstein's spaces.