On compact Riemannian manifolds with noncompact holonomy groups
The notion of holonomy groups of Riemannian manifolds first appeared in the work of E.~Cartan, who used them as a tool in his classification of symmetric spaces. Over the years, several mathematicians (such as Ambrose and Singer, Berger and Simons, to name only a few) contributed to their study. By the middle of the eighties, most problems concerning holonomy groups were solved. (We refer to Chapter~10 in [\textit{A. Besse}, ``Einstein manifolds'', Ergeb. Math. Grenzgeb. 3. Folge 10, Springer, Berlin (1987, Zbl 0613.53001)] for the state of affairs in~1987, and essentially still in~2001.) One major question remained open: is the holonomy group of a compact Riemannian manifold compact? The present article gives the first fully satisfying answer to this question. The author presents the first explicit examples of compact Riemannian manifolds with noncompact holonomy group and then proceeds to examine the structure of such manifolds. To do this, he defines the holonomy representation of the fundamental group of a Riemannian manifold and he shows that a compact Riemannian manifold has compact holonomy group if and only if the image of the fundamental group under this representation is finite. Moreover, he describes all possible holonomy representations on the algebraic level, deducing from this the existence of many compact Riemannian spaces with rather exotic holonomy representations. As a consequence of his detailed study of holonomy representations, he derives the main structure theorem: a finite cover of a compact \(n\)-dimensional Riemannian manifold with noncompact holonomy group is the total space of a bundle over some compact \(b\)-dimensional space, \(b\leq n-4\), whose fibers are flat tori.
scientific article; zbMATH DE number 1547116