On holonomy of Weyl connections in Lorentzian signature
Introduction
The holonomy group of a connection is an important invariant. This motivates the classification problem for holonomy groups. There are classification results for some cases of linear connections. There is a classification of irreducible connected holonomy groups of linear torsion-free connections [27]. Important result is the classification of connected holonomy groups of Riemannian manifolds [6], [7], [10], [24]. Lorentzian holonomy groups are classified [4], [26], [13], [3], [21], [16], [17]. There are partial results for holonomy groups of pseudo-Riemannian manifolds of other signatures [8], [9], [5], [11], [20], [19], [18], [15], [23], [30], [31].
Of certain interests are Weyl manifolds , where c is a conformal class of pseudo-Riemannian metrics and ∇ is a torsion-free linear connection preserving c. In the Riemannian signature the connected holonomy groups of such connection are classified [2], [22].
The result of this paper is a complete classification of connected holonomy groups (equivalently, of holonomy algebras) of Weyl connections in the Lorentzian signature.
In Sections 2 we give necessary background on Weyl manifolds and Berger algebras. Berger algebras have the same algebraic properties as the holonomy algebras and they are candidates to the holonomy algebras. In Section 3 the main results of the paper are stated. Theorem 3 gives the classification of the Berger algebras that preserve proper non-degenerate subspaces of the Minkowski space, these algebras correspond to conformal products in the sense of [2]. Then we assume that the algebras do not preserve any proper non-degenerate subspace of the Minkowski space, these algebras are called weakly irreducible. If such an algebra is different from , then it preserves an isotropic line. Theorem 4 gives the classification of such Berger algebras. Theorem 5 states that each obtained Berger algebra is the holonomy algebra of a Weyl connection. The rest of the paper is dedicated to the proofs of the main results. In Section 4 we describe weakly irreducible subalgebras of . In Section 5 we provide the auxiliary results, namely, we found some spaces of algebraic curvature tensors using the first prolongation of Lie algebras representations. In Section 6 we describe the structure of the spaces of curvature tensors for subalgebras of . The proofs of Theorem 3, Theorem 4 are given in Sections 7 and 8 correspondingly. Finally, in Section 9, for each Berger algebra , we construct a Weyl connection ∇ such that the holonomy algebra of ∇ is , this proves Theorem 5.
The author would like to express his sincere gratitude to A. Galaev for useful discussions and suggestions. The author is grateful to the anonymous referee for careful reading of the paper and valuable comments that greatly affected the final appearance of the paper. The work was supported by the project MUNI/A/1160/2020.
Section snippets
Preliminaries
Denote by a conformal manifold, where M is a smooth manifold, and c is a conformal class of pseudo-Riemannian metrics on M. Recall that two metrics g and h are conformally equivalent if and only if , for some . Definition 1 A Weyl connection ∇ on a conformal manifold is a torsion-free linear connection that preserves the conformal class c. The triple is called a Weyl manifold.
Main results
Here we present the classification of the holonomy algebras of the Weyl connections such that , .
First suppose that is irreducible. Then it is obvious that is irreducible as well. Therefore, , since does not have any proper irreducible algebra [29]. Thus, .
Next, let us suppose that preserves a non-degenerate subspace of . Theorem 3 Let , , be a Berger algebra which admits
Weakly irreducible subalgebras of
We will consider the obvious projection which is a homomorphism of the Lie algebras. For a subalgebra , denote by its projection to . It is obvious that is weakly irreducible if and only if is weakly irreducible. It is clear that if is weakly irreducible and is not contained in , then only the following two situations are possible:
- (a)
contains , in this case
- (b)
does not contain and
Auxiliary results
Let be a pseudo-Euclidean space. We will use the standard isomorphism of the -modules:
Definition 5 For a subalgebra its first prolongation is defined as:
We are interested in the first prolongations, since below we will see that they are tightly related to algebraic curvature tensors. It is well known that and
Lemma 1 Let be a subalgebra of satisfying one of the
Algebraic curvature tensors
For a subalgebra consider the following space: defined in [26], see also [14]. The space is called the space of weak curvature tensors of type .
Theorem 7 Every algebraic curvature tensor is uniquely determined by the elements: by the equalities
Proof of Theorem 3
By the assumption of the theorem we have a -invariant decomposition i.e., We will consider two cases.
Case 1. First suppose, that is irreducible. Then, and . Consider the ideal . Since each subalgebra of is reductive, there exists an ideal such that By the definition of the ideal , the projection may be described in the following way.
Proof of Theorem 4
For convenience, we will use the notation of Theorem 7 for the components of an algebraic curvature tensor R. Consider the cases as in Section 4.
Case a. Let , where of the type 1, 2 or 3 from Theorem 2. If is a Berger algebra, then according to Theorem 9, is the holonomy algebra of a Riemannian manifold. Consider the algebraic curvature tensor R, which is defined by the condition and all other elements are zeros. Since , it holds . Since is a
Realization of Berger algebras
In this section we prove Theorem 5. To do this we show that for every Berger algebra obtained above there exists a Weyl connection ∇ such that the holonomy algebra of ∇ is isomorphic to .
Let be a Weyl manifold. Fix a metric . There exists a 1-form ω such that Let be the Levi-Civita connection corresponding to g. It is well-known that the connection ∇ is uniquely defined by g and ω through the formulas In what
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