Preface; 1. Euclidean geometry; 2. Spherical geometry; 3. Triangulations and Euler numbers; 4. Riemannian metrics; 5. Hyperbolic geometry; 6. Smooth embedded surfaces; 7. Geodesics; 8. Abstract surfaces and Gauss-Bonnet.
This 2007 textbook uses examples, exercises, diagrams, and unambiguous proof, to help students make the link between classical and differential geometries.
Pelham Wilson is Professor of Algebraic Geometry in the Department of Pure Mathematics, University of Cambridge. He has been a Fellow of Trinity College since 1981 and has held visiting positions at universities and research institutes worldwide, including Kyoto University and the Max-Planck-Institute for Mathematics in Bonn. Professor Wilson has over 30 years of extensive experience of undergraduate teaching in mathematics, and his research interests include complex algebraic varieties, Calabi-Yau threefolds, mirror symmetry, and special Lagrangian submanifolds.
'… the patient reader will acquire substantial techniques and
methods that are part of differential geometry and along with that,
much, much more. … The book is certainly a welcome addition to the
literature. It is clear to the reviewer the the text is a labour of
love' Mathematical Reviews
'This book paves its way through a number of geometries with a
clear intent: to promote the understanding of the basic concepts in
differential geometry. The classical 2-dimensional geometries such
as Euclidean, spherical and hyperbolic geometry are the first few
steps. The geometry of the torus, triangulations and Euler numbers
are further issues. Every chapter is followed by an assortment of
helpful examples. The chapters gently direct the reader [towards]
Riemannian metrics and further to geodesics and abstract surfaces.
The Theorema Egregium and Gauss-Bonnet Theorem are also given ample
attention. What distinguishes this book from other lecture notes on
elementary differential geometry? The author does not abruptly
define concepts which otherwise never would have come to the
reader's mind. He rather conveys the ideas, subtly initiated in the
former chapters. In this sense the book is remarkably
self-contained. On the other hand the author does not shun detailed
proofs. All these ingredients make for a successful volume.' Johann
Lang, Zentralblatt MATH
'Curved Spaces provides an elegant, sophisticated treatment of
two-dimensional geometries suitable for advanced undergraduates …
Overall, Wilson has provided a monograph that could ably serve for
an excellent undergraduate capstone experience … Highly
recommended.' S.J. Colley, CHOICE
'The book is written in a nice and precise style and explicit
computations and proofs make the book easy to understand. A
detailed and explicit discussion of the main examples of classical
geometries contributes well to a better understanding of later
generalisations. A list of examples at the end of each chapter
helps as well. It is a very good addition to the literature on the
topic and can be very useful for teachers preparing their courses
as well as for students.' EMS Newsletter
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