ABSTRACT
In this chapter we will give a geometric proof, based on the results of previous chapters, of the Simons Holonomy Theorem about holonomy systems [295]. This result implies the well-known Berger Holonomy Theorem [17], as remarked by Simons. Our proof follows [260], uses similar tools that are based on normal holonomy as those given in [259], and can be regarded as a shorter variation of this. This chapter is reasonably self-contained and there will be only a small overlap with the previous chapters. This is for the convenience of readers who are not particularly interested in submanifold geometry. The only nonstandard fact used in the proof of the Simons Holonomy Theorem is the apparently innocent Corollary 8.2.3. The proof of this result requires the theory developed in previous chapters.
