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1 From Triangles to Manifolds 1

1.1 Geometry 1

1.2 Triangles 1

1.3 Curves in the plane;rotation index and regular homotopy 2

1.4 Euclidean three-space 4

1.5 From coordinate spaces to manifolds 7

1.6 Manifolds;local tools 8

1.7 Homology 9

1.8 Vector fields and generalizations 11

1.9 Elliptic differential equations 12

1.10 Euler characteristic as a source of global invariants 13

1.11 Gauge field theory 13

1.12 Concluding remarks 14

2 Topics in Differential Geometry 17

2.1 General notions on differentiable manifolds 17

2.1.1 Homology and cohomology groups of an abstract complex 17

2.1.2 Product theory 20

2.1.3 An example 22

2.1.4 Algebra of a vector space 24

2.1.5 Differentiable manifolds 26

2.1.6 Multiple integrals 28

2.2 Riemannian manifolds 31

2.2.1 Riemannian manifolds in Euclidean space 31

2.2.2 Imbedding and rigidity problems in Euclidean space 35

2.2.3 Affine connection and absolute differentiation 39

2.2.4 Riemannian metric 41

2.2.5 The Gauss-Bonnet formula 44

2.3 Theory of connections 46

2.3.1 Resume on fiber bundles 47

2.3.2 Connections 49

2.3.3 Local theory of connections;the curvature tensor 52

2.3.4 The homomorphism h and its independence of connection 54

2.3.5 The homomorphism h for the universal bundle 57

2.3.6 The fundamental theorem 60

2.4 Bundles with the classical groups as structural groups 61

2.4.1 Homology groups of Grassmann manifolds 62

2.4.2 Differential forms in Grassmann manifolds 67

2.4.3 Multiplicative properties of the cohomology ring of a Grassmann manifold 74

2.4.4 Some applications 78

2.4.5 Duality theorems 83

2.4.6 An application to projective differential geometry 86

3 Curves and Surfaces in Euclidean Space 89

3.1 Theorem of turning tangents 89

3.2 The four-vertex theorem 94

3.3 Isoperimetric inequality for plane curves 96

3.4 Total curvature of a space curve 100

3.5 Deformation of a space curve 105

3.6 The Gauss-Bonnet formula 108

3.7 Uniqueness theorems of Cohn-Vossen and Minkowski 114

3.8 Bernstein's theorem on minimal surfaces 119

4 Minimal Submanifolds in a Riemannian Manifold 123

4.1 Review of Riemannian geometry 123

4.2 The first vairiation 128

4.3 Minimal submanifolds in Euclidean space 129

4.4 Minimal surfaces in Euclidean space 134

4.5 Minimal submanifolds on the sphere 141

4.6 Laplacian of the second fundamental form 146

4.7 Inequality of Simons 148

4.8 The second variation 151

4.9 Minimal cones in Euclidean space 155

5 Characteristic Classes and Characteristic Forms 163

5.1 Stiefel-Whitney and Pontrjagin classes 163

5.2 Characteristic classes in terms of curvature 165

5.3 Transgression 166

5.4 Holomorphic line bundles and the Nevanlinna theory 168

6 Geometry and Physics 171

6.1 Euclid 171

6.2 Geometry and physics 172

6.3 Groups of transformations 172

6.4 Riemannian geometry 173

6.5 Relativity 173

6.6 Unified field theory 173

6.7 Weyl's abelian gauge field theory 174

6.8 Vector bundles 175

6.9 Why Gauge theory 175

7 The Geometry of G-Structures 177

7.1 Introduction 177

7.2 Riemannian structure 179

7.3 Connections 182

7.4 G-structure 185

7.5 Harmonic forms 189

7.6 Leaved structure 192

7.7 Complex structure 194

7.8 Sheaves 198

7.9 Characteristic classes 202

7.10 Riemann-Roch,Hirzebruch,Grothendieck,and Atiyah-Singer Theorems 207

7.11 Holomorphic mappings of complex analytic manifolds 212

7.12 Isometric mappings of Riemannian manifolds 216

7.13 General theory of G-structures 218