A famous Swiss professor gave a student's course in Basel on Riemann surfaces. After a couple of lectures, a student asked him, "Professor, you have as yet not given an exact de nition of a Riemann surface." The professor answered, "With Riemann surfaces, the main thing is to UNDERSTAND them, not to de ne them." The student's objection was reasonable. From a formal viewpoint, it is of course necessary to start as soon as possible with strict de nitions, but the professor's - swer also has a substantial background. The pure de nition of a Riemann surfaceโ as a complex 1-dimensional complex analytic manifoldโcontributes little to a true understanding. It takes a long time to really be familiar with what a Riemann s- face is. This example is typical for the objects of global analysisโmanifolds with str- tures. There are complex concrete de nitions but these do not automatically explain what they really are, what we can do with them, which operations they really admit, how rigid they are. Hence, there arises the natural questionโhow to attain a deeper understanding? One well-known way to gain an understanding is through underpinning the d- nitions, theorems and constructions with hierarchies of examples, counterexamples and exercises. Their choice, construction and logical order is for any teacher in global analysis an interesting, important and fun creating task
CONTENT
Differentiable manifolds -- Tensor Fields and Differential Forms -- Integration on Manifolds -- Lie Groups -- Fibre Bundles -- Riemannian Geometry -- Some Definitions and Theorems -- Some Formulas and Tables -- Erratum to: Foreword
