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 surface- as a complex 1-dimensional complex analytic manifold-contributes 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 analysis-manifolds 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 question-how 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.
Rezensionen / Stimmen
"This book contains detailed solutions to 375 core exercises on differentiable manifolds, Lie groups, fiber bundles, and Riemannian geometry. It should be helpful to anyone who needs to aquire a solid working technique in these fields and to students who have problems to illustrate concepts, methods, and theorems by examples. The notation is compatible with most standard textbooks, so that teachers might well refer to this book as a reference. The authors summarize the critical definitions and statements where incompatibility might occur. There is a very useful collection of formulas and tables at the end of the book."
(Thomas Püttmann, Bochum, Zentralblatt Math, 2010)
`
I am convinced that this book, which is moreover very nicely edited, will not only be appreciated by students and teachers, but also by many researchers who are interested in the fascinating field of differential geometry and its applications, and who seek a deeper insight into the matter and a better acquaintance with its basic techniques.
'
Professor Frans Cantrijn
in
Mathematical Reviews (2002)
`
This workbook is a really successful attempt to submit to the reader by very clever composed series of exercises and examples covering the whole area of manifolds, Lie groups, fibre bundles, and Riemannian geometry a deep understanding and feeling. The choice and order of the examples and exercises will be extraordinarily helpful and useful for any student or teacher of manifolds and differential geometry.
'
from the
Prologue
by
Professor Jürgen Eichhorn
Reihe
Sprache
Verlagsort
Zielgruppe
Für höhere Schule und Studium
Illustrationen
44
44 s/w Abbildungen
illustrations
Maße
Höhe: 23.5 cm
Breite: 15.5 cm
Dicke: 25 mm
Gewicht
ISBN-13
978-1-4020-0027-0 (9781402000270)
DOI
10.1007/978-90-481-3564-6
Schweitzer Klassifikation
Prologue. Preface. Acknowledgments. About the Authors. 1. Differentiable Manifolds. 2. Tensor Fields and Differential Forms. 3. Integration on Manifolds. 4. Lie Groups. 5. Fibre Bundles. 6. Riemannian Geometry. 7. Some Definitions and Theorems. 8. Some Formulas and Tables. 9**. Manual of `Superficies'. 10. Indices and Notations. References. List of Notations. List of Figures. Index. Ad 9** A CD-ROM is included with the application `Superficies' by Professor A. Montesinos.
