Curved Spaces

From Classical Geometries to Elementary Differential Geometry
Erscheinungsjahr 2011
ISBN: 978-0-521-88629-1
Verlag: Cambridge University Press

Buch, Englisch, 196 Seiten, Format (B × H): 175 mm x 250 mm, Gewicht: 530 g

This self-contained textbook presents an exposition of the well-known classical two-dimensional geometries, such as Euclidean, spherical, hyperbolic, and the locally Euclidean torus, and introduces the basic concepts of Euler numbers for topological triangulations, and Riemannian metrics. The careful discussion of these classical examples provides students with an introduction to the more general theory of curved spaces developed later in the book, as represented by embedded surfaces in Euclidean 3-space, and their generalization to abstract surfaces equipped with Riemannian metrics. Themes running throughout include those of geodesic curves, polygonal approximations to triangulations, Gaussian curvature, and the link to topology provided by the Gauss-Bonnet theorem. Numerous diagrams help bring the key points to life and helpful examples and exercises are included to aid understanding. Throughout the emphasis is placed on explicit proofs, making this text ideal for any student with a basic background in analysis and algebra.

Autoren/Hrsg.

Wilson, P. M. H.

Fachgebiete

• Mathematik | Informatik Mathematik Geometrie Differentialgeometrie

Inhaltsverzeichnis

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.

Über Autor(innen)

Wilson, P. M. H.
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.