On the geometrical properties of hypercomplex four-dimensional Lorentzian Lie groups

In this paper we first classify left-invariant generalized Ricci solitons on four-dimensional hypercomplex Lie groups equipped with three families of left-invariant Lorentzian metrics. Then, on these Lorentzian spaces, we explicitly calculate the energy of an arbitrary left-invariant vector field X and determine the exact form of all left-invariant unit time-like vector fields which are spatially harmonic. Furthermore, we give a complete and explicit description of all homogeneous structures on these spaces in both Riemannian and Lorentzian cases and determine some of their types. The existence of Einstein four-dimensional hypercomplex Lorentzian Lie groups is proved and it is shown that although the results concerning Einstein-like metrics, conformally flatness and some equations in the Riemannian case are much richer than their Lorentzian analogues, in the Lorentzian case, there exist some new critical points of energy functionals, homogeneous structures and geodesic vectors which do not exist in the Riemannian case.

On weakly conformally symmetric pseudo-Riemannian manifolds

In this paper, we study the properties of weakly conformally symmetric pseudo- Riemannian manifolds focusing particularly on the [Formula: see text]-dimensional Lorentzian case. First, we provide a new proof of an important result found in literature; then several new others are stated. We provide a decomposition for the conformal curvature tensor in [Formula: see text]. Moreover, some important identities involving two particular covectors are stated; for example, it is proven that under certain conditions the Ricci tensor and other tensors are Weyl compatible. Topological properties involving the vanishing of the first Pontryagin form are then stated. Further, we study weakly conformally symmetric [Formula: see text]-dimensional Lorentzian manifolds (space-times): it is proven that one of the previously defined co-vectors is null and unique up to a scaling. Moreover, it is shown that under certain conditions, the same vector is an eigenvector of the Ricci tensor and its integral curves are geodesics. Finally, it is stated that such space-time is of Petrov type N with respect to the same vector.

CONSISTENT DISCRETIZATION AND CANONICAL, CLASSICAL AND QUANTUM REGGE CALCULUS

We apply the "consistent discretization" technique to the Regge action for (Euclidean and Lorentzian) general relativity in arbitrary number of dimensions. The result is a well-defined canonical theory that is free of constraints and where the dynamics is implemented as a canonical transformation. In the Lorentzian case, the framework appears to be naturally free of the "spikes" that plague traditional formulations. It also provides a well-defined recipe for determining the integration measure for quantum Regge calculus.

SPACE-TIME VARIABLE SUPERSTRING VACUA (CALABI-YAU COSMIC YARN)

In a general superstring vacuum configuration, the “internal” space (sector) varies in space-time. When this variation is nontrivial only in two spacelike dimensions, the vacuum contains static cosmic strings with finite energy per unit length and which is, up to interactions with matter, an easily computed topological invariant. The total space-time is smooth although the “internal” space is singular at the center of each cosmic string. In a similar analysis of the Wick-rotated Euclidean model, these cosmic strings acquire expected self-interactions. Also, a possibility emerges to define a global time in order to rotate back to the Lorentzian case.