Magnetic reconnection has traditionally been associated exclusively with the presence of magnetic null points or field lines tangential to a boundary. However, in many cases introducing a three‐dimensional perturbation in a two‐and‐half‐dimensional magnetic configuration implies the disappearance of separatrices. Faced with this structural instability of separatrices when going from two‐and‐half to three‐dimensional configurations, several approaches have been investigated to replace the topological ideas familiar in two‐dimensional, but no unanimity has yet emerged on the way reconnection should be defined. While it is true that the field line linkage is continous in three‐dimensional, we show here that extremely thin layers (called quasi‐separatrix layers (QSLs)) are present. In these layers the gradient of the mapping of field lines from one part of a boundary to another is very much larger than normal (by many orders of magnitude). Even for highly conductive media these extremely thin layers behave physically like separatrices. Thus reconnection without null points can occur in QSLs with a breakdown of ideal MHD and a change in connectivity of plasma elements. We have analyzed several twisted flux tube configurations, going progressively from two‐and‐half to three‐dimensional, showing that QSLs are structurally stable features (in contrast to separatrices). The relative thickness w of QSLs depends mainly on the maximum twist; typically, with two turns, w ≈ 10−6, while with four turns, w ≈ 10−12. In these twisted configurations the shape of the QSLs, at the intersection with the lower planar boundary, is typical of the two ribbons observed in two‐ribbon solar flares, confirming that the accompanying prominence eruption involves the reconnection of twisted magnetic structures. We conclude that reconnection occurs in three‐dimensional in thin layers or QSLs, which generalise the traditional separatrices (related only to magnetic null points or field lines tangential to the boundary).

A study is made of the possible holonomy group types of a spacetime for which the energy-momentum tensor corresponds to a null or non-null electromagnetic field, a perfect fluid or a massive scalar field. The case of an Einstein space is also included. The techniques developed are also applied to vacuum and conformally flat space-times and contrasted with already known results in these two cases. Examples are given.

Projective collineations in spacetimes, i.e. vector fields generating local groups of geodesic-preserving diffeomorphisms, are studied. The situation for Einstein spaces is resolved completely and some general results are established regarding arbitrary spacetimes. Examples of proper projective collineations are constructed.

This paper discusses the extent to which one can determine the spacetime metric from a knowledge of a certain subset of the (unparametrised) geodesics of its Levi-Civita connection, that is, from the experimental evidence of the equivalence principle. It is shown that, if the space-time concerned is known to be vacuum, then the Levi-Civita connection is uniquely determined and its associated metric is uniquely determined up to a choice of units of measurement, by the specification of these geodesics. It is further demonstrated that if two space-times share the same unparametrised geodesics and only one is assumed vacuum then their Levi-Civita connections are again equal (and so the other metric is also a vacuum metric) and the first result above is recovered.

There has been some recent interest in the relation between two spacetimes which have the same geodesic paths, that is, spacetimes which are projectively equivalent (sometimes called geodesically equivalent). This paper presents a short and accessible proof of the theorem that if two spacetimes have the same geodesic paths and one of them is an Einstein space then (either each is of constant curvature or) their Lévi-Civitá connections are identical. It also clarifies the relationship between their associated metrics. The results are extended to include the signatures (+ + ++) and (− − ++), and some examples and discussion are given in the case of dimension n > 4. Some remarks are also made which show how these results may be useful in the study of projective symmetry.