Elementary Differential Geometry (Springer Undergraduate Mathematics Series)

✍ Scribed by A.N. Pressley

• Publisher: Springer
• Year: 2010
• Tongue: English
• Leaves: 486
• Edition: 2nd ed. 2010
• Category: Library

✦ Synopsis

Elementary Differential Geometry presents the main results in the differential geometry of curves and surfaces suitable for a first course on the subject. Prerequisites are kept to an absolute minimum – nothing beyond first courses in linear algebra and multivariable calculus – and the most direct and straightforward approach is used throughout.

New features of this revised and expanded second edition include:

a chapter on non-Euclidean geometry, a subject that is of great importance in the history of mathematics and crucial in many modern developments. The main results can be reached easily and quickly by making use of the results and techniques developed earlier in the book.

• Coverage of topics such as: parallel transport and its applications; map colouring; holonomy and Gaussian curvature.

• Around 200 additional exercises, and a full solutions manual for instructors, available via www.springer.com

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✦ Table of Contents

184882890X
Elementary
Differential Geometry
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 first 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 Spherical geometry
7. Curvature of surfaces
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 Poincaré disc model
11.4 Hyperbolic parallels
11.5 Beltrami--Klein model
12. Minimal 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. Möbius transformations
Hints to selected exercises
Solutions
Index