Lectures on Differential Geometry

✍ Scribed by Iskander A. Taimanov

• Publisher: European Mathematical Society
• Year: 2008
• Tongue: English
• Leaves: 221
• Series: Ems Series of Lectures in Mathematics
• Edition: 7
• Category: Library

✦ Synopsis

Differential geometry studies geometrical objects using analytical methods. Like modern analysis itself, differential geometry originates in classical mechanics. For instance, geodesics and minimal surfaces are defined via variational principles and the curvature of a curve is easily interpreted as the acceleration with respect to the path length parameter. Modern differential geometry in its turn strongly contributed to modern physics. This book gives an introduction to the basics of differential geometry, keeping in mind the natural origin of many geometrical quantities, as well as the applications of differential geometry and its methods to other sciences. The text is divided into three parts. The first part covers the basics of curves and surfaces, while the second part is designed as an introduction to smooth manifolds and Riemannian geometry. In particular, Chapter 5 contains short introductions to hyperbolic geometry and geometrical principles of special relativity theory. Here, only a basic knowledge of algebra, calculus and ordinary differential equations is required. The third part is more advanced and introduces into matrix Lie groups and Lie algebras the representation theory of groups, symplectic and Poisson geometry, and applications of complex analysis in surface theory. The book is based on lectures the author held regularly at Novosibirsk State University. It is addressed to students as well as anyone who wants to learn the basics of differential geometry.

✦ Table of Contents

Cover......Page 1
EMS Series of Lectures in Mathematics......Page 3
Lectures on Differential Geometry......Page 4
ISBN 978-3-03719-050-0......Page 5
Preface......Page 6
Contents......Page 8
Part I Curves and surfaces......Page 10
Basic notions of the theory of curves......Page 12
Plane curves......Page 14
Curves in three-dimensional space......Page 17
The orthogonal group......Page 19
Metrics on regular surfaces......Page 24
Curvature of a curve on a surface......Page 26
Gaussian curvature......Page 29
Derivational equations and Bonnet's theorem......Page 31
The Gauss theorem......Page 36
Covariant derivative and geodesics......Page 37
The Euler–Lagrange equations......Page 41
The Gauss–Bonnet formula......Page 47
Minimal surfaces......Page 53
Part II Riemannian geometry......Page 56
Topological spaces......Page 58
Smooth manifolds and maps......Page 60
Tensors......Page 67
Action of maps on tensors......Page 72
Embedding of smooth manifolds into the Euclidean space......Page 76
Metric tensor......Page 78
Affine connection and covariant derivative......Page 79
Riemannian connections......Page 83
Curvature......Page 86
Geodesics......Page 92
The Lobachevskii plane......Page 98
Pseudo-Euclidean spaces and their applications in physics......Page 104
Part III Supplement chapters......Page 110
Conformal parameterization of surfaces......Page 112
The theory of surfaces in terms of the conformal parameter......Page 116
The Weierstrass representation......Page 120
Linear Lie groups......Page 126
Lie algebras......Page 133
Geometry of the simplest linear groups......Page 138
The basic notions of representation theory......Page 144
Representations of finite groups......Page 149
On representations of Lie groups......Page 156
The Poisson bracket and Hamilton's equations......Page 163
Lagrangian formalism......Page 172
Examples of Poisson manifolds......Page 175
Darboux's theorem and Liouville's theorem......Page 179
Hamilton's variational principle......Page 186
Reduction of the order of the system......Page 189
Euler's equations......Page 199
Integrable Hamiltonian systems......Page 203
Bibliography......Page 214
Index......Page 216
Back Cover......Page 221