Vol. 20, No. 2, 1967

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Holonomy groups of indefinite metrics

Hung-Hsi Wu

Vol. 20 (1967), No. 2, 351–392

This paper studies the holonomy group of a riemannian manifold whose metric is allowed to have arbitrary signature; it is meant to supplement the works of Borel, Lichnerowicz and Berger on riemannian manifolds with positive definite metric. We first show that each such holonomy group can be decomposed into the direct product of a finite number of weakly irreducible subgroups of the pseudo-orthogonal group. Those weakly irreducible subgroups which are not irreducible (in the usual sense) we call S W irreducible. So our investigation is reduced to that of these S W irreducible holonomy groups. We actually construct a large class of symmetric spaces with S W irreducible holonomy groups and for the nonsymmetric case, we give an indication of their abundant existence. On the other hand, not every S W irreducible group can be realized as a holonomy group; this fact is shown by an explicit example. We then study the closedness question of S W irreducible subgroups in general, and of holonomy groups in particular. It turns out that algebraic holonomy groups (and hence S W irreducible subgroups in general) need not be closed in Gln but that holonomy groups of symmetric riemannian manifolds of any signature are necessarily closed. Sufficient conditions are also given in order that an S W irreducible subgroup be closed. Finally, we produce various counterexamples to show that many facts known to hold in the positive definite case fail when the metric is allowed to be indefinite.

Mathematical Subject Classification
Primary: 53.72
Received: 27 April 1965
Published: 1 February 1967
Hung-Hsi Wu