1 - Introduction

Summary

This book is written for researchers and graduate students in the field of geometric mechanics, especially the theory of systems with symmetries. A wider audience might include differential geometers, algebraic geometers and singularity theorists. The aim of the book is to show that differential geometry in the sense of Sikorski is a powerful tool for the study of the geometry of spaces with singularities. We show that this understanding of differential geometry gives a complete description of the stratification structure of the space of orbits of a proper action of a connected Lie group G on a manifold P. We also show that the same approach can handle intersection singularities; see Section 8.2.

We assume here that the reader has a working knowledge of differential geometry and the topology of manifolds, and we use theorems in these fields freely without giving proofs or references. On the other hand, the material on differential spaces is developed from scratch. The results on differential spaces are proved in detail. This should make the book accessible to graduate students.

The book is split into two parts. In Part I, we introduce the reader to the differential geometry of singular spaces and prove some results, which are used in Part II to investigate concrete systems. The technique of differential geometry presented here is fairly straightforward, and the reader might get a false impression that the scope of the theory does not differ much from that of the geometry of manifolds.