Einstein equations and Cauchy-Riemann geometry

In Chapter 2 (Sections 2.2-2.5) of this thesis we consider the pure radiation Einstein equations Rμv = if!kμkv, with kμ being a vector field tangent to the twisting congruence of shear-free and null geodesics. We formulate them as invariant equations on the underlying nondegenerate CR-structure (Section 2.4). When we restrict ourselves to the CR-structures realized as hypersurfaces in C2 then the equations reduce to only one real second order equation (Section 2.4) which becomes linear if the so called NUT parameter [32) of the solution vanishes. In this case (Section 2.5) the equation can be solved under certain assumptions. As an example we present solutions to the system with vanishing NUT parameter for CRstructures admitting 3-dimensional symmetry groups of Bianchi types V h. The metrics corresponding to these solutions admit no symmetries in general. Most of them are new. Another application of our formulation of the pure radiation Einstein equations is a general solution which admits at least three infinitesimal conformal symmetries (Section 2.3). A short Section 2.6 gives CR-invariant formulation of the twisting type N vacuum Einstein equations. These equations will be studied elsewhere. The last Section of Chapter 2 provides first explicit solutions describing purely radiative Einstein-Maxwell fields with twisting shear-free geodesic null congruences; also first examples of Einstein-Maxwell fields with twisting congruences which possess both radiative and charged part are presented there. Chapter 3 deals with orthogonal almost complex structures over a 4-dimensional Euclidean manifold. In four dimensions such structures are Euclidean analogs of null congruences as described in the last section of the Introduction. In this analogy the integrability conditions for the almost complex structure to be complex correspond to the geodesic and shear-free property of the null congruence [21). In Sections 3.1, 3.2 we define an Euclidean analog of the Newman-Penrose formalism [31). As an application of the formalism we discuss integrability conditions for a local existence of orthogonal complex structure on Euclidean 4-manifolds (Section 3.4). In this section it is also showed which orthogonal complex structures are admitted by conformally flat metrics. This gives a part of an Euclidean analog of the Kerr theorem. An Euclidean analog of the Goldberg-Sachs theorem [13) is presented in (Section 3.5). It follows from this theorem and Section 3.4 that an Euclidean 4-manifold satisfying the Einstein equations Rμv = Agμv admits orthogonal complex structure if its metric is algebraically special in the sense of an Euclidean analog of the Cartan-PetrovPenrose classification given in Section 3.3. Such a classification was known in the spinorial language. Our classification refers to the properties of orthogonal almost complex structures. The results of the whole Chapter 3 seem to be new.