微分幾何基礎

Andrew Pressley （A.普雷斯利,英國）是國際知名學者，在數學界享有盛譽。本書凝聚了作者多年科研和教學成果，適用於科研工作者、高校教師和研究生。 ^[1] 

微分幾何基礎講述的是曲線和平面的微分幾何學的主要結論適合於本科生個學期的課程。在改版中有如下新的特徵：有一章專門講述非歐幾何，該課題在數學史上具有重要的影響且對現代數學發展的影響也至關重要；書中包括的課題有：平行移動及其應用、地圖設色、完整的高斯曲率。讀者對象：數學專業本科生及相關科研工作者。 ^[1] 

Preface

Contents

1．Curves in the plane and in space

1．1 What is a curve？

1．2 Arc-length

1．3 Reparametrization

1．4 Closed curves

1．5 Level curves versus parametrized curves

2．How much does a curve curve？

2．1 Curvature

2．2 Plane curves

2．3 Space curves

3．Global properties of curves

3．1 Simple closed curves

3．2 The isoperimetric inequality

3．3 The four vertex theorem

4．Surfaces in three dimensions

4．1 What is a surface？

4．2 Smooth surfaces

4．3 Smooth maps

4．4 Tangents and derivatives

4．5 Normals and orientability

5．Examples of surfaces

5．1 Level surfaces

5．2 Quadric surfaces

5．3 Ruled surfaces and surfaces of revolution

5．4 Compact surfaces

5．5 Triply orthogonal systems

5．6 Applications of the inverse function theorem

6．The flrst fundamental form

6．1 Lengths of curves on surfaces

6．2 Isometries of surfaces

6．3 Conformal mappings of surfaces

6．4 Equiareal maps and a theorem of Archimedes

6．5 Sphericalgeometry

7．Curvature of 8urfaces

7．1 The second fundamental form

7．2 The Gauss and Weingarten maps

7．3 Normal and geodesic curvatures

7．4 Parallel transport and covariant derivative

8．Gaussian， mean and principal curvatures

8．1 Gaussian and mean curvatures

8．2 Principal curvatures of a surface

8．3 Surfaces of constant Gaussian curvature

8．4 Flat surfaces

8．5 Surfaces of constant mean curvature

8．6 Gaussian curvature of compact surfaces

9．Geodesics

9．1 Definition and basic properties

9．2 Geodesic equations

9．3 Geodesics on surfaces of revolution

9．4 Geodesics as shortest paths

9．5 Geodesic coordinates

10．Gauss' Theorema Egregium

10．1 The Gauss and Codazzi-Mainardi equations

10．2 Gauss' remarkable theorem

10．3 Surfaces of constant Gaussian curvature

10．4 Geodesic mappings

11．Hyperbolic geometry

11．1 Upper half-plane model

11．2 Isometries of H

11．3 Poincare disc model

11．4 Hyperbolic parallels

11．5 Beltrami-Klein model

12．Minmal surfaces

12．1 Plateau's problem

12．2 Examples of minimal surfaces

12．3 Gauss map of a minimal surface

12．4 Conformal parametrization of minimal surfaces

12．5 Minimal surfaces and holomorphic functions

13．The Gauss-Bonnet theorem

13．1 Gauss-Bonnet for simple closed curves

13．2 Gauss-Bonnet for curvilinear polygons

13．3 Integration on compact surfaces

13．4 Gauss-Bonnet for compact surfaces

13．5 Map colouring

13．6 Holonomy and Gaussian curvature

13．7 Singularities of vector fields

13．8 Critical points

A0．Inner product spaces and self-adjoint linear maps

A1．Isometries of Euclidean spaces

A2．Mobius transformations

Hints to selected exercises

Solutions

Index ^[2]