We formulate an approach to the geometry of Riemann–Cartan spaces provided with nonholonomic distributions defined by generic off-diagonal and nonsymmetric metrics inducing effective nonlinear and affine connections. Such geometries can be modelled by moving nonholonomic frames on (pseudo) Riemannian manifolds and describe various types of nonholonomic Einstein, Eisenhart–Moffat and Finsler–Lag...

Beginning with the self-dual two-forms approach to the Einstein equations, we show how, by choosing basis spinors which are proportional to solutions of the Dirac equation, we may rewrite the vacuum Einstein equations in terms of a set of divergence-free vector fields, which obey a particular set of chiral equations. Upon imposing the Jacobi identity upon these vector fields, we reproduce a pre...

We exhibit several families of Jacobi–Videv pseudo-Riemannian manifolds which are not Einstein. We also exhibit Jacobi–Videv algebraic curvature tensors where the Ricci operator defines an almost complex structure.

This paper is a sequel of [8]. In algebraic geometry, when discussing the compactification of the moduli space of complex manifold X with ample canonical bundle KX , it is necessary to consider holomorphic degeneration family π : X → B, where Xt = π(t) are smooth for t 6= 0, X and X0 are QGorenstein, such that the canonical bundle of Xt for t 6= 0 and the dualizing sheaf of X0 are ample. We wil...

The structure of the general, inhomogeneous solution of (bosonic) Einstein-matter systems in the vicinity of a cosmological singularity is considered. We review the proof (based on ideas of BelinskiiKhalatnikov-Lifshitz and technically simplified by the use of the ArnowittDeser-Misner Hamiltonian formalism) that the asymptotic behaviour, as one approaches the singularity, of the general solutio...

The averaging problem in cosmology and the approach of macroscopic gravity to resolve the problem is discussed. The averaged Einstein equations of macroscopic gravity are modified on cosmological scales by the macroscopic gravitational correlation tensor terms as compared with the Einstein equations of general relativity. This correlation tensor satisfies a system of structure and field equatio...