This book is based on lecture courses given by the authors over the past decade and a half to various student audiences, most of them final year undergraduates or beginning graduates. It is meant particularly for those who wish to study relativity theory or classical mechanics from a geometrical viewpoint. In each of these subjects one can go quite far without knowing about differentiable manifolds, and the arrangement of the book exploits this. The essential ideas are first introduced in the context of affine space; this is enough for special relativity and vectorial mechanics. Then manifolds are introduced and the essential ideas are suitably adapted; this makes it possible to go on to general relativity and canonical mechanics. The book ends with some chapters on bundles and connections which may be useful in the study of gauge fields and such matters. The “applicability” of the material appears in the choice of examples, and sometimes in the stating of conditions which may not always be the strongest ones and in the omission of some proofs which we feel add little to an understanding of results.

We have included a great many exercises. They range from straightforward verifications to substantial practical calculations. The end of an exercise is marked with the sign □. Exercises are numbered consecutively in each chapter; where we make reference to an exercise, or indeed a section, in a chapter other than the current one we do so in the form “Exercise n of Chapter p” or something of that kind.