Variational problems in geometry

Contents\0......Page 6
Preface to the English Edition\0......Page 8
Preface\0......Page 10
Outlines and Objectives of the Theory\0......Page 14
1.1. Are length and energy of curves\0......Page 20
1.2. Euler\'s equation\0......Page 28
1.3. Connections and covariant differentiation\0......Page 35
1.4. Geodesics\0......Page 47
1.5. Minimal length property of geodesics\0......Page 57
Summary\0......Page 62
Exercises\0......Page 63
2.1. The first variation formula\0......Page 66
2.2. Curvature tensor\0......Page 73
2.3. The second variation formula\0......Page 84
2.4. Existence of minimal geodesics\0......Page 88
2.5. Applications to Riemannian geometry\0......Page 96
Exercises\0......Page 101
3.1. Energy of maps\0......Page 104
3.2. Tension fields\0......Page 109
3.3. The first variation formula\0......Page 118
3.4. Harmonic maps\0......Page 122
3.5. The second variation formula\0......Page 129
Exercise\0......Page 133
4.1. The heat flow method\0......Page 138
4.2. Existence of local time-dependent solutions\0......Page 149
4.3. Existence of global time-dependent solutions\0......Page 160
4.4. Existence and uniqueness of harmonic maps\0......Page 169
4.5. Applications to R.iemannian geometry\0......Page 174
Exercises\0......Page 179
A. I. Fundamentals of manifolds\0......Page 182
A.2. Fundamentals of functional analysis\0......Page 192
Prospects for Contemporary Mathematics\0......Page 202
Solutions to Exercise Problems\0......Page 206
Bibliography\0......Page 220
Index\0......Page 226