On compact Riemannian manifolds with noncompact holonomy groups

Abstract

Solving a long standing problem in Riemannian geometry we construct a compact Riemannian manifold with a noncompact holonomy group.As the title indicates we then prove structure theorems for these manifolds.We employ an argument of Cheeger and Gromoll [1971] to show that the holonomy group of a compact Riemannian manifold is compact if and only if the image of the so called holonomy representation of its fundamental group is finite.Then we characterize these holonomy representations algebraically.As a consequence we prove that a finite cover of a compact Riemannian manifold M' n ) with a noncompact holonomy group is the total space of a torus bundle over another compact Riemannian manifold ß' 6 ' with b < n -4.