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**On the classification of Lorentzian holonomy groups.**
*(English)*
Zbl 1129.53029

By the Wu-de Rham decomposition theorem (1964), the classification of holonomy groups of simply connected pseudo-Riemannian manifolds reduces to that of indecomposable such groups. Let \(V\) be a pseudo-Euclidian vector space. A subgroup \(H \subset O(V)\) is called indecomposable if all proper \(H\)-invariant subspaces \(U\subset V\) are degenerate. Irreducible subgroups \(H\subset O(V)\) are indecomposable but for indefinite scalar products there exist also reducible indecomposable groups. Irreducible holonomy groups of simply connected pseudo-Riemannian manifolds have been classifed by Berger (1955). For Lorentzian signature, \(\text{SO}_0(V)\) is the only such group.

Let \(V\) be a Lorentzian vector space and \(H\subset O(V)\cong O(n+1,1)\) a reducible indecomposable subgroup. Such groups have been classified by L. Bérard-Bergery and A. Ikemakhen [On the holonomy of Lorentzian manifolds, Proc. Symp. Pure Math. 54, 27–40 (1993; Zbl 0807.53014)]. Obviously \(H\) preserves a degenerate subspace \(U\subset V\) and hence the null line \(L=U\cap U^\perp\). The stabilizer \(O(V)_L\) of \(L\) in the pseudo-orthogonal group \(O(V)\) is isomorphic to the group of similarities of the Euclidian space \(\mathbb{R}^n\). Thus, \(H\) can be considered as a subgroup of \((\mathbb{R}^+\times O(n))\ltimes\mathbb{R}^n\). Let us denote by \(G\) the \(O(n)\)-projection of \(H\). The main result of the paper is that \(G\subset O(n)\) is the holonomy group of a Riemannian manifold if \(H\subset O(n+1,1)\) is the holonomy group of a Lorentzian manifold. The proof relies on the representation theory of compact groups and is rather involved. This finishes the classification of holonomy groups of simply connected Lorentzian manifolds, since A. S. Galaev [Int. J. Geom. Methods Mod. Phys. 3, No. 5-6, 1025–1045 (2006; Zbl 1112.53039)] has constructed a Lorentzian metric with holonomy group \(H\) for any connected reducible indecomposable subgroup \(H\subset O(n+1,1)\) for which the \(O(n)\)-projection \(G\) is a Riemannian holonomy group.

Before the work of Galaev such metrics were known for certain classes but not for all of the indecomposable subgroups \(H\subset O(n+1,1)\), by the work of Cahen and Wallach (1970), who classified simply connected Lorentzian symmetric spaces, and Bérard-Bergery and Ikemakhen, who constructed the first non-symmetric examples. The classification of holonomy groups of simply connected pseudo-Riemannian manifolds of index 2 is still open but Galaev has classified holonomy groups of simply connected pseudo-Kähler manifolds of index 2 in his thesis [Holonomy groups and special geometric structures of pseudo-Kählerian manifolds of index 2. Humboldt-University Berlin (2006; Zbl 1127.53305)]. Unfortunately, the analogue of Leistner’s result about the \(O(n)\)-projection does not hold, which makes the latter classification surprisingly complicated.

Let \(V\) be a Lorentzian vector space and \(H\subset O(V)\cong O(n+1,1)\) a reducible indecomposable subgroup. Such groups have been classified by L. Bérard-Bergery and A. Ikemakhen [On the holonomy of Lorentzian manifolds, Proc. Symp. Pure Math. 54, 27–40 (1993; Zbl 0807.53014)]. Obviously \(H\) preserves a degenerate subspace \(U\subset V\) and hence the null line \(L=U\cap U^\perp\). The stabilizer \(O(V)_L\) of \(L\) in the pseudo-orthogonal group \(O(V)\) is isomorphic to the group of similarities of the Euclidian space \(\mathbb{R}^n\). Thus, \(H\) can be considered as a subgroup of \((\mathbb{R}^+\times O(n))\ltimes\mathbb{R}^n\). Let us denote by \(G\) the \(O(n)\)-projection of \(H\). The main result of the paper is that \(G\subset O(n)\) is the holonomy group of a Riemannian manifold if \(H\subset O(n+1,1)\) is the holonomy group of a Lorentzian manifold. The proof relies on the representation theory of compact groups and is rather involved. This finishes the classification of holonomy groups of simply connected Lorentzian manifolds, since A. S. Galaev [Int. J. Geom. Methods Mod. Phys. 3, No. 5-6, 1025–1045 (2006; Zbl 1112.53039)] has constructed a Lorentzian metric with holonomy group \(H\) for any connected reducible indecomposable subgroup \(H\subset O(n+1,1)\) for which the \(O(n)\)-projection \(G\) is a Riemannian holonomy group.

Before the work of Galaev such metrics were known for certain classes but not for all of the indecomposable subgroups \(H\subset O(n+1,1)\), by the work of Cahen and Wallach (1970), who classified simply connected Lorentzian symmetric spaces, and Bérard-Bergery and Ikemakhen, who constructed the first non-symmetric examples. The classification of holonomy groups of simply connected pseudo-Riemannian manifolds of index 2 is still open but Galaev has classified holonomy groups of simply connected pseudo-Kähler manifolds of index 2 in his thesis [Holonomy groups and special geometric structures of pseudo-Kählerian manifolds of index 2. Humboldt-University Berlin (2006; Zbl 1127.53305)]. Unfortunately, the analogue of Leistner’s result about the \(O(n)\)-projection does not hold, which makes the latter classification surprisingly complicated.

Reviewer: Vicente Cortés (Hamburg)