arXiv subjects
Publications and source records attributed to Pawel Nurowski.
We discuss the unique existence, arising by analogy to that in algebraically special space-times, of a CR structure realized on null infinity for any asymptotically flat Einstein or Einstein-Maxwell space-time.
We consider the nonstandard inclusion of SO(3) in SO(5) associated with a 5-dimensional irreducible representation. The tensor $Υ$ representing this reduction is found to be given by a ternary symmetric form with special properties. A 5-dimensional manifold $(M,g,Υ)$ with Riemannian metric $g$ and ternary form generated by such a tensor has a corresponding SO(3) structure, whose Gray-Hervella type classification is established using so(3)-valued connections with torsion. Structures with antisymmetric torsions, we call them the nearly integrable SO(3) structures, are studied in detail. In particular, it is shown that the integrable models (those with vanishing torsion) are isometric to the symmetric spaces $M_+= SU(3)/SO(3)$, $M_-=SL(3,R)/SO(3)$, $M_0=R^5$. We also find all nearly integrable SO(3) structures with transitive symmetry groups of dimension $d>5$ and some examples for which $d=5$. Given an SO(3) structure $(M,g,Υ)$, we define its "twistor space" T to be the $S^2$-bundle of those unit 2-forms on $M$ which span $R^3=so(3)$. The 7-dimensional twistor manifold T is then naturally equipped with several CR and $G_2$ structures. The ensuing integrability conditions are discussed and interpreted in terms of the Gray-Hervella type classification.
Cartan's list of 3-dimensional Weyl structures with reduced holonomy is revisited. We show that the only Einstein-Weyl structures on this list correspond to the structures generated by the solutions of the dKP equation.
We construct a family of split signature Einstein metrics in four dimensions, corresponding to particular classes of third order ODEs considered modulo fiber preserving transformations of variables.
We construct polynomial conformal invariants, the vanishing of which is necessary and sufficient for an $n$-dimensional suitably generic (pseudo-)Riemannian manifold to be conformal to an Einstein manifold. We also construct invariants which give necessary and sufficient conditions for a metric to be conformally related to a metric with vanishing Cotton tensor. One set of invariants we derive generalises the set of invariants in dimension four obtained by Kozameh, Newman and Tod. For the conformally Einstein problem, another set of invariants we construct gives necessary and sufficient conditions for a wider class of metrics than covered by the invariants recently presented by M. Listing. We also show that there is an alternative characterisation of conformally Einstein metrics based on the tractor connection associated with the normal conformal Cartan bundle. This plays a key role in constructing some of the invariants. Also using this we can interpret the previously known invariants geometrically in the tractor setting and relate some of them to the curvature of the Fefferman-Graham ambient metric.
The equivalence problem for second order ODEs given modulo point transformations is solved in full analogy with the equivalence problem of nondegenerate 3-dimensional CR structures. This approach enables an analog of the Feffereman metrics to be defined. The conformal class of these (split signature) metrics is well defined by each point equivalence class of second order ODEs. Its conformal curvature is interpreted in terms of the basic point invariants of the corresponding class of ODEs.
Necessary and sufficient conditions for a space-time to be conformal to an Einstein space-time are interpreted in terms of curvature restrictions for the corresponding Cartan conformal connection.
We show that every 2nd order ODE defines a 4-parameter family of projective connections on its 2-dimensional solution space. In a special case of ODEs, for which a certain point transformation invariant vanishes, we find that this family of connections always has a preferred representative. This preferred representative turns out to be identical to the projective connection described in Cartan's classic paper "Sur les Varietes a Connection Projective".
A Lorentzian manifold is defined here as a smooth pseudo-Riemannian manifold with a metric tensor of signature ((2n +1, 1)). A Robinson manifold is a Lorentzian manifold (M) of dimension (\geqslant 4) with a subbundle (N) of the complexification of (TM) such that the fibers of (N\to M) are maximal totally null (isotropic) and ([\Sec N, \Sec N]\subset \Sec N). Robinson manifolds are close analogs of the proper Riemannian, Hermite manifolds. In dimension 4, they correspond to space-times of general relativity, foliated by a family of null geodesics without shear. Such space-times, introduced in the 1950s by Ivor Robinson, played an important role in the study of solutions of Einstein's equations: plane and sphere-fronted waves, the Gödel universe, the Kerr solution, and their generalizations, are among them. In this survey article, the analogies between Hermite and Robinson manifolds are presented in considerable detail.
The geometry of P, the bundle of null directions over an Einstein space-time, is studied. The full set of invariants of the natural G-structure on P is constructed using the Cartan method of equivalence. This leads to an extension of P which is an elliptic fibration over the space-time. Examples are given which show that such an extension, although natural, is not unique. A reinterpretation of the Petrov classification in terms of the fibres of an extension of P is presented.
We define a certain differential system on an open set of $R^6$. The system locally defines a Lorentzian 4-manifold satisfying the Einstein equations. The converse statement is indicated and its details are postponed to the furthcoming paper.
Given a conformally nonflat Einstein spacetime we define a fibration $P$ over it. The fibres of this fibration are elliptic curves (2-dimensional tori) or their degenerate counterparts. Their topology depends on the algebraic type of the Weyl tensor of the Einstein metric. The fibration $P$ is a double branched cover of the bundle of null direction over the spacetime and is equipped with six linearly independent 1-forms which satisfy certain relatively simple system of equations.