Lectures on differential geometry

✍ Scribed by Taimanov I.

• Publisher: EMS
• Year: 2008
• Tongue: English
• Leaves: 219
• Series: Ems Series of Lectures in Mathematics
• 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

Preface......Page 5
I Curves and surfaces......Page 9
Basic notions of the theory of curves......Page 11
Plane curves......Page 13
Curves in three-dimensional space......Page 16
The orthogonal group......Page 18
Metrics on regular surfaces......Page 23
Curvature of a curve on a surface......Page 25
Gaussian curvature......Page 28
Derivational equations and Bonnet's theorem......Page 30
The Gauss theorem......Page 35
Covariant derivative and geodesics......Page 36
The Euler–Lagrange equations......Page 40
The Gauss–Bonnet formula......Page 46
Minimal surfaces......Page 52
II Riemannian geometry......Page 55
Topological spaces......Page 57
Smooth manifolds and maps......Page 59
Tensors......Page 66
Action of maps on tensors......Page 71
Embedding of smooth manifolds into the Euclidean space......Page 75
Metric tensor......Page 77
Affine connection and covariant derivative......Page 78
Riemannian connections......Page 82
Curvature......Page 85
Geodesics......Page 91
The Lobachevskii plane......Page 97
Pseudo-Euclidean spaces and their applications in physics......Page 103
III Supplement chapters......Page 109
Conformal parameterization of surfaces......Page 111
The theory of surfaces in terms of the conformal parameter......Page 115
The Weierstrass representation......Page 119
Linear Lie groups......Page 125
Lie algebras......Page 132
Geometry of the simplest linear groups......Page 137
The basic notions of representation theory......Page 143
Representations of finite groups......Page 148
On representations of Lie groups......Page 155
The Poisson bracket and Hamilton's equations......Page 162
Lagrangian formalism......Page 171
Examples of Poisson manifolds......Page 174
Darboux's theorem and Liouville's theorem......Page 178
Hamilton's variational principle......Page 185
Reduction of the order of the system......Page 188
Euler's equations......Page 198
Integrable Hamiltonian systems......Page 202
Bibliography......Page 213
Index......Page 215