前辅文
Part I What is Geometry and Differential Geometry
1 WhatIsGeometry?.
1.1 Geometry as a logical system
1.2 Coordinatization of space
1.3 Space based on the group concept
1.4 Localization of geometry
1.5 Globalization
1.6 Connections in a fiber bundle
1.7 An application to biology
1.8 Conclusion
2 Differential Geometry
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
2.3.2 Euler-Poincar´e characteristic.
2.3.3 Minimal submanifolds
2.3.4 Isometricmappings
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 LectureV
3.6 LectureVI
3.7 LectureVII
3.8 LectureVIII
Part III DifferentiableManifolds
4 Multilinear Algebra
4.1 The tensor (or Kronecker) product
4.2 Tensor spaces
4.3 Symmetry and skew-symmetry
4.4 Duality in exterior algebra
4.5 Inner product
5 DifferentiableManifolds
5.1 Definition of a differentiable manifold
5.2 Tangent space
5.3 Tensor bundles
5.4 Submanifolds
6 Exterior Differential Forms
6.1 Exterior differentiation
6.2 Differential systems
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 parallelismof Levi-Civita
8.2 Sectional curvature
8.3 Normal coordinates
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 -Osserman theorem
10.4 Inequality on Gaussian curvature.
11 Pseudospherical Surface
11.1 General theorems.
11.2 B¨acklund’s theorem.
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还没人写过短评呢
还没人写过短评呢