Holonomy groups of compact Riemannian manifolds: A classification in dimensions up to ten

The possible holonomy groups of compact, locally irreducible Riemannian manifolds are studied. Motivated by applications arising in physics, it is not required that these manifolds be simply connected. In this paper a complete classification up to dimension ten is given, with particular emphasis on the manifolds of dimension six or less. In this latter case, there exist actual examples for every class, so that the possible holonomy groups can be characterized exactly.