文档介绍：This page intentionally left blank
Curved Spaces
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– 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.
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
plex algebraic varieties, Calabi–Yau threefolds, mirror symmetry and special
Lagrangian submanifolds.
Curved Spaces
From Classical Geometries to
Elementary Differential Geometry
P. M. H. Wilson
Department of Pure Mathematics, University of Cambridge,
and Trinity College, Cambridge
CAMBRIDGE UNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Buildin