Topics in differential geometry （英文版）

前辅文
1 From Triangles to Manifolds
1.1 Geometry
1.2 Triangles
1.3 Curves in the plane
1.4 Euclidean three-space
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前辅文
1 From Triangles to Manifolds
1.1 Geometry
1.2 Triangles
1.3 Curves in the plane
1.4 Euclidean three-space
1.5 From coordinate spaces to manifolds
1.6 Manifolds
1.7 Homology
1.8 Vector fields and generalizations
1.9 Elliptic differential equations
1.10 Euler characteristic as a source of global invariants
1.11 Gauge field theory
1.12 Concluding remarks
2 Topics in Differential Geometry
2.1 General notions on differentiablemanifolds
2.1.1 Homology and cohomology groups of an abstract complex
2.1.2 Product theory
2.1.3 An example
2.1.4 Algebra of a vector space
2.1.5 Differentiablemanifolds
2.1.6 Multiple integrals
2.2 Riemannianmanifolds
2.2.1 Riemannianmanifolds in Euclidean space
2.2.2 Imbedding and rigidity problems in Euclidean space
2.2.3 Affine connection and absolute differentiation
2.2.4 Riemannianmetric
2.2.5 The Gauss-Bonnet formula
2.3 Theory of connections
2.3.1 Resume on fiber bundles
2.3.2 Connections
2.3.3 Local theory of connections
2.3.4 The homomorphism h and its independence of connection
2.3.5 The homomorphism h for the universal bundle
2.3.6 The fundamental theorem
2.4 Bundles with the classical groups as structural groups
2.4.1 Homology groups of Grassmann manifolds
2.4.2 Differential forms in Grassmannmanifolds
2.4.3 Multiplicative properties of the cohomology ring of a Grassmannmanifold
2.4.4 Some applications
2.4.5 Duality theorems
2.4.6 An application to projective differential geometry
3 Curves and Surfaces in Euclidean Space
3.1 Theoremof turning tangents
3.2 The four-vertex theorem
3.3 Isoperimetric inequality for plane curves
3.4 Total curvature of a space curve
3.5 Deformation of a space curve
3.6 The Gauss-Bonnet formula
3.7 Uniqueness theorems of Cohn-Vossen and Minkowski
3.8 Bernstein’s theorem on minimal surfaces
4 Minimal Submanifolds in a Riemannian Manifold
4.1 Review of Riemannian geometry
4.2 The first variation
4.3 Minimal submanifolds in Euclidean space
4.4 Minimal surfaces in Euclidean space
4.5 Minimal submanifolds on the sphere
4.6 Laplacian of the second fundamental form
4.7 Inequality of Simons
4.8 The second variation
4.9 Minimal cones in Euclidean space
5 Characteristic Classes and Characteristic Forms
5.1 Stiefel-Whitney and Pontrjagin classes
5.2 Characteristic classes in terms of curvature
5.3 Transgression
5.4 Holomorphic line bundles and the Nevanlinna theory
6 GeometryandPhysics
6.1 Euclid
6.2 Geometry and physics
6.3 Groups of transformations
6.4 Riemannian geometry
6.5 Relativity
6.6 Unified field theory
6.7 Weyl’s abelian gauge field theory
6.8 Vector bundles
6.9 Why Gauge theory
7 The Geometry of G-Structures
7.1 Introduction
7.2 Riemannian structure
7.3 Connections
7.4 G-structure
7.5 Harmonic forms
7.6 Leaved structure
7.7 Complex structure
7.8 Sheaves
7.9 Characteristic classes
7.10 Riemann-Roch, Hirzebruch, Grothendieck, and Atiyah-Singer Theorems
7.11 Holomorphic mappings of complex analyticmanifolds
7.12 Isometricmappings of Riemannianmanifolds
7.13 General theory of G-structures
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