Differential Geometry and Integral Geometry

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Author: Shiing-Shen Chern
Language: English
ISBN/ISSN: 9787040465181
Published on: 2016-10
Hardcover

Part Ⅰ What is Geometry and Differential Geometry
1 What Is Geometry?
1.1 Geometry as a logical system; Euclid
1.2 Coordinatization of space; Descartes
1.3 Space based on the group concept; Klein's Erlanger Programm
1.4 Localization of geometry; Gauss and Riemann
1.5 Globalization; topology
1.6 Connections in a fiber bundle; Elie Cartan
1.7 An application to biology
1.8 Conclusion
2 Differential Geometry; Its Past and Its Future
2.1 Introduction
2.2 The development of some fundamental notions and tools
2.3 Formulation of some problems with discussion of related results
2.3.1 Riemannian manifolds whose sectional curvatures keep a constant sign
2.3.2 Euler—Poincare characteristic
2.3.3 Minimal submanifolds
2.3.4 Isometric mappings
2.3.5 Holomorphic mappings
Part Ⅱ Lectures on Integral Geometry
3 Lectures on Integral Geometry
3.1 Lecture Ⅰ
3.1.1 Buffon's needle problem
3.1.2 Bertrand's parabox
3.2 Lecture Ⅱ
3.3 Lecture Ⅲ
3.4 Lecture Ⅳ
3.5 Lecture Ⅴ
3.6 Lecture Ⅵ
3.7 Lecture Ⅶ
3.8 Lecture Ⅷ
Part Ⅲ Differentiable Manifolds
4 Multilinear Algebra
4.1 The tensor (or Kronecker) product
4.2 Tensor spaces
4.3 Symmetry and skew—symmetry; Exterior algebra
4.4 Duality in exterior algebra
4.5 Inner product
5 Differentiable Manifolds
5.1 Definition of a differentiable manifold
5.2 Tangent space
5.3 Tensor bundles
5.4 Submanifolds; Imbedding of compact manifolds
6 Exterior Differential Forms
6.1 Exterior differentiation
6.2 Differential systems; Frobenius's theorem
6.3 Derivations and anti—derivations
6.4 Infinitesimal transformation
6.5 Integration of differential forms
6.6 Formula of Stokes
7 Affine Connections
7.1 Definition of an affine connection: covariant differential
7.2 The principal bundle
7.3 Groups of holonomy
7.4 Affine normal coordinates
8 Riemannian Manifolds
8.1 The parallelism of Levi—Civita
8.2 Sectional curvature
8.3 Normal coordinates; Existence of convex neighbourhoods
8.4 Gauss—Bonnet formula
8.5 Completeness
8.6 Manifolds of constant curvature
Part Ⅳ Lecture Notes on Differentiable Geometry
9 Review of Surface Theory
9.1 Introduction
9.2 Moving frames
9.3 The connection form
9.4 The complex structure
10 Minimal Surfaces
10.1 General theorems
10.2 Examples
10.3 Bernstein —Osserrnan theorem
10.4 Inequality on Gaussian curvature
11 Pseudospherical Surface
11.1 General theorems
11.2 Backlund's theorem 



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