ABSTRACT
A connection on a Riemannian manifold M can be interpreted as a tool for comparing different tangent spaces by means of parallel transport. In general, parallel transport depends upon the curve that is chosen for joining two given points. This dependence is measured by the so-called holonomy group; that is, the group of linear isometries of a tangent space TpM generated by all parallel transports along loops based at p. Holonomy plays an important role in Riemannian geometry, in particular in the context of special structures on manifolds, for example, Ka¨hler, hyperka¨hler, or quaternionic Ka¨hler structures. Holonomy is a concept that can be defined for any connection on a vector bundle. In this chapter we will study the holonomy group of the normal connection of a submanifold, the so-called normal holonomy group of a submanifold. The purpose of this chapter is to explain how the theory of holonomy can be used to study submanifold geometry.
