Symmetries and Curvature Structure in General Relativity

Projective structure and holonomy in four-dimensional Lorentz manifolds

“…Let B m denote the range space of f at m (suitably interpreted according to the agreed identification of bivectors) so that dim B m equals the curvature rank at m and is ≤ 6. This leads to a convenient algebraic classification of Riem at m into five mutually exclusive and disjoint curvature classes (for further details, see [10]). …”

“…A space-time (M, g) (or some subset of it) which has the same curvature class at each point of M is said to be of that class. The subset of M consisting of points at which the curvature class is A is an open subset of M ( [10], p 393) and the analogous subset arising from the class O is closed (and has empty interior in the manifold topology of M if (M, g) is non-flat). It is important in what is to follow that the equation R abcd k d = 0 at m has no non-trivial solutions for k ∈ T m M if the curvature class at m is A or B, a unique independent solution (the vector r above) if the curvature class at m is C and two independent solutions if the curvature class at m is D (and which span the blade of * F in (3)).…”

“…If dim B m ≥ 4 the curvature class at m is necessarily A. The following result will be useful in what is to follow and the details and proof can be found in [10]. Theorem 1.…”

“…It is a consequence of (8) that the blade of F (if F is simple) and each of the canonical pair of blades of F (if F is non-simple) are eigenspaces of h with respect to g, that is, for F simple, any k ∈ T m M in the blade of F satisfies h ab k b = ωg ab k b where the eigenvalue ω ∈ R is independent of k, and similarly for each of the canonical blades if F is non-simple (but with possibly different eigenvalues for these blades) [10].…”

“…If (M, g) is of curvature class A, (7) givesg = φg for some (smooth) function φ on M and the Bianchi identity may be used to show that φ is constant on M [10]. Further, if the condition on g and g of having equal curvatures is weakened tog ab;[cd] = 0 on M , where a semi-colon denotes a ∇-covariant derivative, the Ricci identity forg shows that the condition of theorem 1 is satisfied forg = h on M .…”