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On the holonomy of Lorentzian manifolds. (English) Zbl 0807.53014
Greene, Robert (ed.) et al., Differential geometry. Part 2: Geometry in mathematical physics and related topics. Proceedings of a summer research institute, held at the University of California, Los Angeles, CA, USA, July 8-28, 1990. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 54, Part 2, 27-40 (1993).
Holonomy and restricted holonomy groups of Riemannian manifolds have been studied since E. Cartan introduced them and they are now well-understood. In particular, the complete list of possible (restricted) holonomy groups is known. This is in sharp contrast to the pseudo-Riemannian case where the corresponding list is still unknown. This more difficult situation comes from the fact that in this case one has to consider the notion of (strictly) irreducibility and also the notion of (strictly) indecomposability. Taking into account the result of Berger and Cahen- Wallach one knows the list of restricted holonomy groups for connected strictly irreducible nonlocally symmetric pseudo-Riemannian manifolds, the classification of irreducible pseudo-Riemannian symmetric spaces and that of the strictly indecomposable, nonstrictly irreducible locally symmetric Lorentz manifolds. Hence, using the generalizations given by Wu of the theorem of Borel, Lichnerowicz and de Rham, we are left, in the Lorentzian case, with that of strictly indecomposable, nonstrictly irreducible, nonlocally symmetric Lorentz manifolds. In this paper the authors make some contribution to the theory for this class. First, they give a complete list of connected subgroups of O$$(1,m - 1)$$ $$(m$$ being the dimension of the manifold) which are not irreducible but do not leave invariant any nondegenerate proper subspace. This is still not sufficient to give a solution to the problem of the determination of all possible restricted holonomy groups, but it is a first step because it restricts the possible cases. Then they derive a further obstruction and mention that no further obstructions are known, up to now. Finally, they give examples of Lorentz manifolds whose restricted holonomy groups are not closed. It is shown that their dimension must necessarily be greater than 5. We note that the case of general pseudo-Riemannian manifolds is still more complicated since even the classification of pseudo-Riemannian symmetric spaces is unknown.
For the entire collection see [Zbl 0773.00023].

MSC:
 53B30 Local differential geometry of Lorentz metrics, indefinite metrics 53C35 Differential geometry of symmetric spaces