Holonomy groups in Riemannian geometry (Q2850737)

The main aim of these lecture notes is to provide the general ideas and techniques in the study of holonomy groups of Riemannian manifolds. The content of the book is divided into 5 chapters. After a brief introduction and motivations in Chapter 1, the authors deal with vector bundles and principal bundles, and discuss the way that connections can be defined on them in Chapter 2. Among other things, a large number of examples are given and the existence of the Levi-Civita connection for any metric is proved, from the principal bundle point of view.NEWLINENEWLINEChapter 3 starts with the definitions of parallel transport and holonomy and continues to basic properties of the holonomy groups. In the final part of this chapter the Ambrose-Singer theorem is proved which relates the holonomy of a connection in a principal bundle with the curvature form of the connection.NEWLINENEWLINENEWLINEChapter 4 focusses on Riemannian holonomy. The de Rham decomposition theorem is proved, a principle for splitting a Riemannian manifold into a Cartesian product of Riemannian manifolds by splitting the tangent bundle into irreducible spaces under the action of the local holonomy groups, and the classification theorem of Berger concerning the possible irreducible holonomies. The last chapter is devoted to the study of each one of the groups in Berger's classification. In the reviewer's opinion, this book is very well written and a valuable contribution to the literature on holonomy groups in Riemannian geometry.