On holonomy of Weyl connections in Lorentzian signature

doi:10.1016/j.difgeo.2021.101759

Differential Geometry and its Applications

Volume 76, June 2021, 101759

https://doi.org/10.1016/j.difgeo.2021.101759 Get rights and content

Abstract

Holonomy algebras of Weyl connections in Lorentzian signature are classified. In particular, examples of Weyl connections with all possible holonomy algebras are constructed.

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 $(M, c, \nabla)$ , 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 $co (1, n + 1)$ , 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 $co (1, n + 1)$ . 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 $co (1, n + 1)$ . The proofs of Theorem 3, Theorem 4 are given in Sections 7 and 8 correspondingly. Finally, in Section 9, for each Berger algebra $g \subset co (1, n + 1)$ , we construct a Weyl connection ∇ such that the holonomy algebra of ∇ is $g$ , 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 $(M, c)$ 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 $h = e^{2 f} g$ , for some $f \in C^{\infty} (M)$ .

Definition 1

A Weyl connection ∇ on a conformal manifold $(M, c)$ is a torsion-free linear connection that preserves the conformal class c. The triple $(M, c, \nabla)$ is called a Weyl manifold.

By preserving a conformal class, we understand that if

g \in c

, then there exists a 1-form

ω_{g}

such that

\nabla

Main results

Here we present the classification of the holonomy algebras $g \subset co (1, n + 1)$ of the Weyl connections such that $g ⊄ so (1, n + 1)$ , $n ⩾ 0$ .

First suppose that $g \subset co (1, n + 1)$ is irreducible. Then it is obvious that ${pr}_{so (1, n + 1)} g \subset so (1, n + 1)$ is irreducible as well. Therefore, ${pr}_{so (1, n + 1)} g = so (1, n + 1)$ , since $so (1, n + 1)$ does not have any proper irreducible algebra [29]. Thus, $g = co (1, n + 1)$ .

Next, let us suppose that $g$ preserves a non-degenerate subspace of $R^{1, n + 1}$ .

Theorem 3

Let $g \subset co (1, n + 1)$ , $g ⊄ so (1, n + 1)$ , be a Berger algebra which admits

Weakly irreducible subalgebras of $co {(1, n + 1)}_{R p}$

We will consider the obvious projection ${pr}_{so (1, n + 1)} : co (1, n + 1) = R \oplus so (1, n + 1) \to so (1, n + 1),$ which is a homomorphism of the Lie algebras. For a subalgebra $g \subset co (1, n + 1)$ , denote by $\overline{g}$ its projection to $so (1, n + 1)$ . It is obvious that $g$ is weakly irreducible if and only if $\overline{g}$ is weakly irreducible. It is clear that if $g$ is weakly irreducible and is not contained in $so (1, n + 1)$ , then only the following two situations are possible:

(a)
$g$ contains $R {id}_{R^{1, n + 1}}$ , in this case $g = R {id}_{R^{1, n + 1}} \oplus \overline{g};$
(b)
$g$ does not contain $R {id}_{R^{1, n + 1}}$ and

Auxiliary results

Let $R^{r, s}$ be a pseudo-Euclidean space. We will use the standard isomorphism $\land^{2} R^{r, s} ≅ so (r, s)$ of the $so (r, s)$ -modules: $(X \land Y) Z = (X, Z) Y - (Y, Z) X .$

Definition 5

For a subalgebra $g \subset gl (n, R)$ its first prolongation is defined as: $g^{(1)} : = {φ \in Hom (R^{n}, g) | φ (X) Y = φ (Y) X} .$

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 ${(so (r, s))}^{(1)} = 0$ and $co {(r, s)}^{(1)} = {Z \land \cdot + (Z, \cdot) {id}_{R^{r, s}} | Z \in R^{r, s}} ≅ R^{r, s} .$

Lemma 1

Let $h$ be a subalgebra of $so (r, s)$ satisfying one of the

Algebraic curvature tensors

For a subalgebra $h \subset so (n)$ consider the following space: $P (h) = {P \in Hom (R^{n}, h) | g (P (X) Y, Z) + g (P (Y) Z, X) + g (P (Z) X, Y) = 0, X, Y, Z \in R^{n}}$ defined in [26], see also [14]. The space $P (h)$ is called the space of weak curvature tensors of type $h$ .

Theorem 7

Every algebraic curvature tensor $R \in R (co {(1, n + 1)}_{R p})$ is uniquely determined by the elements: $μ, λ \in R, X_{0}, Z_{0} \in R^{n}, γ \in Hom (R^{n}, R), P \in P (so (n)), K \in ⊙^{2} R^{n}, S + τ {id}_{R^{n}} \in R (co (n)), where S \in Hom (\land^{2} R^{n}, so (n)), τ \in Hom (\land^{2} R^{n}, R)$ by the equalities $\begin{matrix} R (p, q) & = (μ, λ, A_{0}, X_{0}), \\ R (p, V) & = ((Z_{0}, V), (Z_{0}, V), Z_{0} \land V, - (A_{0} + μ E_{n}) V), \\ R (U, V) & = ((A_{0} U, V), ( \end{matrix}$

Proof of Theorem 3

By the assumption of the theorem we have a $g$ -invariant decomposition $R^{1, n + 1} = R^{1, k + 1} \oplus R^{n - k},$ i.e., $g \subset so (1, k + 1) \oplus so (n - k) \oplus R {id}_{R^{1, n + 1}} .$ We will consider two cases.

Case 1. First suppose, that ${pr}_{so (1, k + 1)} g$ is irreducible. Then, ${pr}_{so (1, k + 1)} g = so (1, k + 1)$ and $k ⩾ 1$ . Consider the ideal $a = g \cap so (n - k) \subset {pr}_{so (n - k)} g$ . Since each subalgebra of $so (n - k)$ is reductive, there exists an ideal $b \subset {pr}_{so (n - k)} g$ such that ${pr}_{so (n - k)} g = a \oplus b .$ By the definition of the ideal $a$ , the projection ${pr}_{so (1, n + 1)} g$ 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 $g = R {id}_{R^{1, n + 1}} \oplus \overline{g}$ , where $\overline{g}$ of the type 1, 2 or 3 from Theorem 2. If $g$ is a Berger algebra, then according to Theorem 9, $h$ is the holonomy algebra of a Riemannian manifold. Consider the algebraic curvature tensor R, which is defined by the condition $μ = 1$ and all other elements are zeros. Since $R \in R (g)$ , it holds $R {id}_{R^{1, n + 1}} \subset L (R (g))$ . Since $\overline{g}$ is a

Realization of Berger algebras

In this section we prove Theorem 5. To do this we show that for every Berger algebra $g \subset co (1, n + 1)$ obtained above there exists a Weyl connection ∇ such that the holonomy algebra of ∇ is isomorphic to $g$ .

Let $(M, c, \nabla)$ be a Weyl manifold. Fix a metric $g \in c$ . There exists a 1-form ω such that $\nabla g = 2 ω \otimes g .$ Let $\overline{\nabla}$ 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 $\nabla = \overline{\nabla} + K,$ $g (K_{X} (Y), Z) = g (Y, Z) ω (X) + g (X, Z) ω (Y) - g (X, Y) ω (Z) .$ In what

References (32)

N.I. Bezvitnaya
Holonomy algebras of pseudo-hyper-Kählerian manifolds of index 4
Differ. Geom. Appl.
(2013)
A.S. Galaev
The spaces of curvature tensors for holonomy algebras of Lorentzian manifolds
Differ. Geom. Appl.
(2005)
A.S. Galaev
One component of the curvature tensor of a Lorentzian manifold
J. Geom. Phys.
(2010)
W. Ambrose et al.
A theorem on holonomy
Trans. Am. Math. Soc.
(1953)
F. Belgun et al.
Weyl-parallel forms, conformal products and Einstein-Weyl manifolds
Asian J. Math.
(2011)
H. Baum
Holonomy groups of Lorentzian manifolds: a status report
L. Bérard-Bergery et al.
On the holonomy of Lorentzian manifolds
L. Bérard-Bergery et al.
Sur l'holonomie des variétés pseudo-riemanniennes de signature (n,n)
Bull. Soc. Math. Fr.
(1997)
M. Berger
Sur les groupers d'holonomie des variétés àconnexion affine et des variétés riemanniennes
Bull. Soc. Math. Fr.
(1955)
A.L. Besse
Einstein Manifolds
(1987)

N.I. Bezvitnaya

Holonomy groups of pseudo-quaternionic-Kählerian manifolds of non-zero scalar curvature

Ann. Glob. Anal. Geom.

(2011)

R.L. Bryant

Metrics with exceptional holonomy

Ann. Math.

(1987)

A. Fino et al.

Holonomy groups of $G_{2}^{⁎}$ -manifolds

Trans. Am. Math. Soc.

(2019)

A.S. Galaev

Metrics that realize all Lorentzian holonomy algebras

Int. J. Geom. Methods Mod. Phys.

(2006)

A.S. Galaev

Note on the holonomy groups of pseudo-Riemannian manifolds

Math. Notes

(2013)

A.S. Galaev

Holonomy of Einstein Lorentzian manifolds

Class. Quantum Gravity

(2010)

Cited by (3)

Recurrent Lorentzian Weyl spaces
2022, arXiv
Parallel spinors on Lorentzian Weyl spaces
2021, Monatshefte fur Mathematik
Parallel spinors on Lorentzian Weyl spaces
2020, arXiv

View full text

On holonomy of Weyl connections in Lorentzian signature

Abstract

Introduction

Section snippets

Preliminaries

Main results

Weakly irreducible subalgebras of co(1,n+1)Rp

Auxiliary results

Algebraic curvature tensors

Proof of Theorem 3

Proof of Theorem 4

Realization of Berger algebras

Differ. Geom. Appl.

Differ. Geom. Appl.

J. Geom. Phys.

A theorem on holonomy

Trans. Am. Math. Soc.

Weyl-parallel forms, conformal products and Einstein-Weyl manifolds

Asian J. Math.

Holonomy groups of Lorentzian manifolds: a status report

On the holonomy of Lorentzian manifolds

Sur l'holonomie des variétés pseudo-riemanniennes de signature (n,n)

Bull. Soc. Math. Fr.

Sur les groupers d'holonomie des variétés àconnexion affine et des variétés riemanniennes

Bull. Soc. Math. Fr.

Einstein Manifolds

Holonomy groups of pseudo-quaternionic-Kählerian manifolds of non-zero scalar curvature

Ann. Glob. Anal. Geom.

Metrics with exceptional holonomy

Ann. Math.

Holonomy groups of G2⁎-manifolds

Trans. Am. Math. Soc.

Metrics that realize all Lorentzian holonomy algebras

Int. J. Geom. Methods Mod. Phys.

Note on the holonomy groups of pseudo-Riemannian manifolds

Math. Notes

Holonomy of Einstein Lorentzian manifolds

Class. Quantum Gravity

Weakly irreducible subalgebras of $co {(1, n + 1)}_{R p}$

Holonomy groups of $G_{2}^{⁎}$ -manifolds