Lectures on Differential Geometry (Ems Series of Lectures in Mathematics)

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