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Connections with irreducible holonomy representations. (English) Zbl 1037.53035
A subgroup of a linear group is called a Berger group if it satisfies all algebraic conditions for being the holonomy group of a torsion free affine connection. These conditions were introduced by Berger who also produced the list of such groups. Famous problems of differential geometry were to establish which groups from this list are realized by the holonomy groups of such connections and, in particular, which Berger groups are realized by torsion free connections compatible with a Riemannian metric. Now both problems are solved and the final solution to the classification problem for irreducible holonomies of torsion free affine connections was obtained by S. Merkulov and the author [Ann. Math. (2), 150, 77–149 (1999; Zbl 0992.53038); S. A. Heggett (ed.) et al., The geometric universe, Oxford University Press, 395–402 (1998; Zbl 0907.53018)]. It appears that all irreducible Berger groups are realized by the holonomy groups. This paper contains another proof of this classification theorem which is based on the classical representation theory.

MSC:
53C29 Issues of holonomy in differential geometry
53C05 Connections, general theory
53C28 Twistor methods in differential geometry
53C17 Sub-Riemannian geometry
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
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