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On the holonomy of pseudo-Riemannian manifolds with signature \((n,n)\). (Sur l’holonomie des variétés pseudo-riemanniennes de signature \((n,n)\).) (French) Zbl 0916.53033

Let \(M\) be a non-decomposable (into a direct product) pseudo-Riemannian manifold \(M\) of signature \((n,n)\) with a non-irreducible restricted holonomy group \(G\). The authors describe the structure of possible holonomy groups \(G\) under the assumption that it preserves a decomposition of the tangent space \(V= T_xM\) into a direct sum of two isotropic subspaces. Moreover, for \(n=2\), the list of all possible holonomy groups \(G\) is obtained. Some examples of pseudo-Riemannian 4-manifolds with different holonomy groups \(G\) from this list are presented. The special cases of pseudo-Kähler manifolds and symmetric spaces are discussed.

MSC:

53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53C05 Connections (general theory)
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