Holonomy and Projective Equivalence in 4-Dimensional Lorentz Manifolds

Abstract

Abstract. A study is made of 4-dimensional Lorentz manifolds which are projectively related, that is, whose Levi-Civita connections give rise to the same (unparameterised) geodesics. A brief review of some relevant recent work is provided and a list of new results connecting projective relatedness and the holonomy type of the Lorentz manifold in question is given. This necessitates a review of the possible holonomy groups for such manifolds which, in turn, requires a certain convenient classification of the associated curvature tensors. These reviews are provided.

Projective structure and holonomy in four-dimensional Lorentz manifolds

“…This can be done quite efficiently using the holonomy theory described in section 3 and allows for easier access to them directly through the curvature tensor. First the following technical lemma is required and which was given in [23].…”

“…The proof can mostly be found in [23] except for the last part. This follows by noting that, from (19)(a), if c = 0, λ is covariantly constant on U and either vanishes everywhere or nowhere on U .…”

“…Now apply the Ricci identity to a and use (15) to get (a ab;cd − a ab;dc =)a ae R e bcd + a be R e acd = g ac λ bd + g bc λ ad − g ad λ bc − g bd λ ac (18) where λ ab = λ a;b (= λ ba ). This leads to the following lemma which is (mostly) a special case of a more detailed result in [3,23] and for which a definition is required. Suppose m ∈ M , that F ∈ Λ m M and that the curvature tensor Riem of (M, g) satisfies R ab cd F cd = αF ab (α ∈ R) at m, so that F may be regarded as a (real) eigenvector of the map f in (1) (recalling the liberties taken in identifying bivectors described in section 1).…”

“…It follows that all projectively related metrics g ′ together with their associated 1-forms ψ will be found if all pairs (a, λ) can be found satisfying (15) and with a non-degenerate. [In the above definition of g ′ (≡ F a −1 ) it is pointed out that, in [23], a typographical error arose which resulted in g ′ being erroneously written as g ′ = e 2χ a(= F a). ]…”

“…Now the holonomy algebra of (U, g) is a subalgebra of that of (M, g) and it follows from table 1 that for each holonomy algebra listed in part (i), its subalgebras are also contained in this list (excluding, of course, the R 5 subalgebra of R 12 which cannot occur as the holonomy algebra of a space-time). Thus one may proceed with a hierarchical proof, using theorems 2, 3 and 5 in [23] applied to the restricted space-time (U, g). It follows from this that, in each case, λ = 0 on U .…”