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Classical, exceptional, and exotic holonomies: A status report. (English) Zbl 0882.53014
Besse, Arthur L. (ed.), Actes de la table ronde de géométrie différentielle en l’honneur de Marcel Berger, Luminy, France, 12–18 juillet, 1992. Paris: Société Mathématique de France. Sémin. Congr. 1, 93-165 (1996).
This is a lucid report, written by a leading expert of the subject, dealing with “the status of the problem of determining the groups that can occur as the irreducible holonomy of a torsion-free affine connection on some manifold”.
The problem of prescribed holonomy, i.e., which subgroups \(H\) of \(GL(V)\), where \(V\) is an \(n\)-dimensional vector space, can be considered as the holonomy (groups) of some torsion-free connection on some \(n\)-dimensional manifold \(M\) was systematically treated by M. Berger. In his fundamental paper [Bull. Soc. Math. France 83, 279-330 (1955; Zbl 0068.36002)], M. Berger gave a list of the possible holonomies, claiming also that it was complete up to a finite number of missing terms. This work triggered a lot of research since then. However, the classification of non-metric holonomies turned out to be incomplete and it was proved that exotic holonomies, representing missing entries from Berger’s original list, do exist.
The report starts with a paragraph devoted to the holonomy and \(G\)-structures along with the necessary algebraic machinery needed in the formulation of Berger’s criteria, under which \(H\) can occur as the holonomy of a torsion-free connection. It is also explained why the general problem can be handled by a classification of torsion-free \(H\)-structures, which in turn can be treated by methods of Cartan-Kähler theory.
Paragraph 2 contains Berger’s list related with the metric cases, mainly the possible irreducible holonomies for pseudo-Riemannian metrics which are not locally symmetric. Each entry of the list is reviewed individually and there is an analysis of the particular problem involved. There are also included references of recent contributions related with certain structures and the question whether they do or do not satisfy Berger’s criteria (some of these structures belong to the so-called exceptional cases).
The non-metric case, i.e., the possible irreducible holonomies for affine connections which are not locally symmetric and do not preserve any nonzero quadratic form, are treated in Paragraph 3. Again there is an extensive and detailed analysis of the entries of Berger’s corresponding list (with some modifications), grouped in certain categories (affine, conformal, symplectic, etc.), as well as information about the relevant connections and their space.
Paragraph 4 is a short account of exotic holonomies, discovered in the late ‘80s. As mentioned earlier, they correspond to irreducibly acting groups which can occur as holonomy groups but are not included in Berger’s general list. According to the author, the full classification of the possible exotic examples is far from complete.
Summarizing, the report is a very informative presentation of the development and mathematics involved in the subject of prescribed holonomy. The reader, especially the non-expert, will learn a lot from it.
For the entire collection see [Zbl 0859.00016].

53B05 Linear and affine connections
53C10 \(G\)-structures
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
58A15 Exterior differential systems (Cartan theory)