微分几何专题PDF|Epub|txt|kindle电子书版本网盘下载

微分几何专题PDF格式电子书版下载

注意：本站所有压缩包均有解压码： 点击下载压缩包解压工具

1 From Triangles to Manifolds1

1.1 Geometry1

1.2 Triangles1

1.3 Curves in the plane;rotation index and regular homotopy2

1.4 Euclidean three-space4

1.5 From coordinate spaces to manifolds7

1.6 Manifolds;local tools8

1.7 Homology9

1.8 Vector fields and generalizations11

1.9 Elliptic differential equations12

1.10 Euler characteristic as a source of global invariants13

1.11 Gauge field theory13

1.12 Concluding remarks14

2 Topics in Differential Geometry17

2.1 General notions on differentiable manifolds17

2.1.1 Homology and cohomology groups of an abstract complex17

2.1.2 Product theory20

2.1.3 An example22

2.1.4 Algebra of a vector space24

2.1.5 Differentiable manifolds26

2.1.6 Multiple integrals28

2.2 Riemannian manifolds31

2.2.1 Riemannian manifolds in Euclidean space31

2.2.2 Imbedding and rigidity problems in Euclidean space35

2.2.3 Affine connection and absolute differentiation39

2.2.4 Riemannian metric41

2.2.5 The Gauss-Bonnet formula44

2.3 Theory of connections46

2.3.1 Resume on fiber bundles47

2.3.2 Connections49

2.3.3 Local theory of connections;the curvature tensor52

2.3.4 The homomorphism h and its independence of connection54

2.3.5 The homomorphism h for the universal bundle57

2.3.6 The fundamental theorem60

2.4 Bundles with the classical groups as structural groups61

2.4.1 Homology groups of Grassmann manifolds62

2.4.2 Differential forms in Grassmann manifolds67

2.4.3 Multiplicative properties of the cohomology ring of a Grassmann manifold74

2.4.4 Some applications78

2.4.5 Duality theorems83

2.4.6 An application to projective differential geometry86

3 Curves and Surfaces in Euclidean Space89

3.1 Theorem of turning tangents89

3.2 The four-vertex theorem94

3.3 Isoperimetric inequality for plane curves96

3.4 Total curvature of a space curve100

3.5 Deformation of a space curve105

3.6 The Gauss-Bonnet formula108

3.7 Uniqueness theorems of Cohn-Vossen and Minkowski114

3.8 Bernstein's theorem on minimal surfaces119

4 Minimal Submanifolds in a Riemannian Manifold123

4.1 Review of Riemannian geometry123

4.2 The first vairiation128

4.3 Minimal submanifolds in Euclidean space129

4.4 Minimal surfaces in Euclidean space134

4.5 Minimal submanifolds on the sphere141

4.6 Laplacian of the second fundamental form146

4.7 Inequality of Simons148

4.8 The second variation151

4.9 Minimal cones in Euclidean space155

5 Characteristic Classes and Characteristic Forms163

5.1 Stiefel-Whitney and Pontrjagin classes163

5.2 Characteristic classes in terms of curvature165

5.3 Transgression166

5.4 Holomorphic line bundles and the Nevanlinna theory168

6 Geometry and Physics171

6.1 Euclid171

6.2 Geometry and physics172

6.3 Groups of transformations172

6.4 Riemannian geometry173

6.5 Relativity173

6.6 Unified field theory173

6.7 Weyl's abelian gauge field theory174

6.8 Vector bundles175

6.9 Why Gauge theory175

7 The Geometry of G-Structures177

7.1 Introduction177

7.2 Riemannian structure179

7.3 Connections182

7.4 G-structure185

7.5 Harmonic forms189

7.6 Leaved structure192

7.7 Complex structure194

7.8 Sheaves198

7.9 Characteristic classes202

7.10 Riemann-Roch,Hirzebruch,Grothendieck,and Atiyah-Singer Theorems207

7.11 Holomorphic mappings of complex analytic manifolds212

7.12 Isometric mappings of Riemannian manifolds216

7.13 General theory of G-structures218