ON THE REAL THEORY OF 4-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)-struc ture, and they satisfy the condition <(E)over bar (alpha)> = E(beta) f or 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 t ensor 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(b eta), respectively. These subtensors satisfy the same conditions as th e bundles E(alpha) and E(beta). Con formally semiflat and flat structu res and their geometrical characteristics are studied. The principal 2 -directions are defined, and conditions for their integrability are ob tained. These investigations for the CO(1,3)-structure are connected w ith Petrov's classification of Einstein's spaces.