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This paper considers 4-dimensional manifolds upon which there is a Lorentz metric, h, and a symmetric connection and which are originally assumed unrelated. It then derives sufficient conditions on the metric and connection (expressed through the curvature tensor) for the connection to be the Levi-Civita connection of some (local) Lorentz metric, g, and calculates the relationship between g and h. Some examples are provided which help to assess the strength of the sufficient conditions derived.

It is shown that the only conformal or projective vector fields admitted by the Reissner-Nordstrom metric are its standard Killing vector fields.

The motivation for this paper is the recent interest in the study of symme tries in general relativity and its purpose is to discuss the mathematical foundations required for such a study. The general (formal and informal) ideas of what constitutes a symmetry of space-time are discussed and developed and the idea of a Lie algebra of symmetry vector fields is studied in detail. The relationship between such Lie algebras and the ideas of Lie transformation group theory (Palais' theorems) is stated and a general theorem regarding the orbits of such symmetries is given. Finally some specific symmetries in general relativity are explored and some of their similarities and differences noted.

Those space-times admitting special conformal vector fields and those admitting special projective vector fields have recently been studied. In this paper these two classes of space times are shown to be very closely related to each other. Certain uniqueness features of (and necessary extra symmetries contained in) the associated Lie algebras are discussed and the dimensionality of each of the algebras is computed.

It is shown that a spacetime admitting symmetries may be disjointly decomposed into an open dense finite union of open sets in each of which the type (spacelike, timelike or null) and dimension of the symmetry orbits and the Petrov type of the Weyl tensor are constant, together with a closed non-empty subset of M which has no interior. It is also shown that if the symmetries admitted are affine (including Killing and homothetic), conformal or projective, the subset of points of M at which the orbit dimension is maximum is open and dense (i.e. `almost everywhere') in M.

A general study is made of conformal vector fields on a four‐dimensional Lorentz manifold with particular emphasis being laid on the structure of the zeros (critical points) of such vectors fields. The implications for general relativity are investigated and a discussion of conformal vector fields in generalized plane wave space‐times is given. An attempt is made to clarify the well‐known theorem of Bilyalov and Defrise‐Carter.

Consider the curvature tensor components Rbcda on a space-time M calculated from given space-time metric components gab. The number of alternative space-time metrics for M which yield the same curvature components is known to be heavily restricted, the exact ambiguity in the metric being dependent on the form of the curvature components. In this paper it is shown that, in spite of these ambiguities, the algebraic structure of the energy-momentum tensor and the Weyl tensor on M are essentially determined.

In this letter some recent results due to Collins and Lang (ibid. vol. 3, p.1143, (1986)) on homotheties and singularities in Einstein's theory are discussed and extended.

A theorem is given which states that, under certain conditions, local symmetries (isometries, homotheties, affine collineations and conformal motions) can all be extended to global symmetries on spacetime.

An intuitive directional symmetry for a null direction with respect to a given local Lorentz observer is suggested. It is shown that the existence of a null direction with such a symmetry with respect to the given Lorentz observer implies that the null direction is a Debever- Penrose direction. A weak converse is also given. In the vacuum case it is shown that the existence of such a null direction is equivalent to the space-time being algebraically special in the Petrov classification and having that null direction as a repeated principal null direction. The results are generalized to non-vacuum space-times. Finally it is suggested how a vacuum space-time might be classified according to the type of symmetries it possesses.

Some similarities between 'P-P' wave space-times and complex recurrent and conformally symmetric space-times in general relativity are discussed.