Einstein-Like Lorentz Metrics and Three-Dimensional Curvature Homogeneity of Order One

We completely classify three-dimensional Lorentz manifolds, curvature homogeneous up to order one, equipped with Einstein-like metrics. New examples arise with respect to both homogeneous examples and three-dimensional Lorentz manifolds admitting a degenerate parallel null line field.

Homotopy classification of line fields and of Lorentz metrics on closed manifolds

The problem of classifying line fields or, equivalently, Lorentz metrics up to homotopy is studied. Complete solutions are obtained in many cases, e.g. for all closed smooth manifolds N, orientable or not, of dimension n ≡ 0(4) and, in particular, in the classical space-time dimension 4.Our approach is based on the singularity method which allows us to classify the monomorphisms u from a given (abstract) line bundle α over N into the tangent bundle. The analysis of the transition to the image line field u(α) then centers around the notion of ‘antipodality’.We express our classification results in terms of standard (co-)homology and characteristic classes. Moreover, we illustrate them for large families of concrete sample manifolds by explicit bijections or by calculating the number of line fields.