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.
