Elementary Differential Geometry / Edition 2

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Curves and surfaces are objects that everyone can see, and many of the questions that can be asked about them are natural and easily understood. Differential geometry is concerned with the precise mathematical formulation of some of these questions. It is a subject that contains some of the most beautiful and profound results in mathematics yet many of these are accessible to higher-level undergraduates.

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

Praise for the first edition:

"The text is nicely illustrated, the definitions are well-motivated and the proofs are particularly well-written and student-friendly…this book would make an excellent text for an undergraduate course, but could also well be used for a reading course, or simply read for pleasure."

Australian Mathematical Society Gazette

"Excellent figures supplement a good account, sprinkled with illustrative examples."

Times Higher Education Supplement

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Editorial Reviews

From the Publisher
From the reviews of the second edition:

“I am very happy to report that the new edition of Pressley’s Elementary Differential Geometry is an even better book than the first edition … . full solutions to all problems given in an appendix. … Most of the problems are in the book and have solutions in the back. … The upshot is that this is still an excellent book and still my first choice for an undergraduate introduction to differential geometry.” (Fernando Q. Gouvêa, The Mathematical Association of America, May, 2010)

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Product Details

  • ISBN-13: 9781848828902
  • Publisher: Springer London
  • Publication date: 3/18/2010
  • Series: Springer Undergraduate Mathematics Series
  • Edition description: 2nd ed. 2012
  • Edition number: 2
  • Pages: 395
  • Sales rank: 785,301
  • Product dimensions: 6.10 (w) x 9.10 (h) x 1.20 (d)

Meet the Author

Andrew Pressley is Professor of Mathematics at King’s College London, UK.

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Table of Contents



1 Curves in the plane and in space

1.1 What is a curve? 1

1.2 Arc-length 9

1.3 Reparametrization 13

1.4 Closed curves 19

1.5 Level curves versus parametrized curves 23

2 How much does a curve curve?

2.1 Curvature 29

2.2 Plane curves 34

2.3 Space curves 46

3 Global properties of curves

3.1 Simple closed curves 55

3.2 The isoperimetric inequality 58

3.3 The four vertex theorem 62

4 Surfaces in three dimensions

4.1 What is a surface? 67

4.2 Smooth surfaces 76

4.3 Smooth maps 82

4.4 Tangents and derivatives 85

4.5 Normals and orientability 89

5 Examples of surfaces

5.1 Level surfaces 95

5.2 Quadric surfaces 97

5.3 Ruled surfaces and surfaces of revolution 104

5.4 Compact surfaces 109

5.5 Triply orthogonal systems 111

5.6 Applications of the inverse function theorem 116

6 The first fundamental form

6.1 Lengths of curves on surfaces 121

6.2 Isometries of surfaces 126

6.3 Conformal mappings of surfaces 133

6.4 Equiareal maps mid a theorem of Archimedes 139

6.5 Spherical geometry 148

7 Curvature of surfaces

7.1 The second fundamental form 159

7.2 The Gauss and Weingarten maps 162

7.3 Normal and geodesic curvatures 165

7.4 Parallel transport and covariant derivative 170

8 Gaussian, mean and principal curvatures

8.1 Gaussian and mean curvatures 179

3.2 Principal curvatures of a surface 187

8.3 Surfaces of constant Gaussian curvature 196

8.4 Flat surfaces 201

8.5 Surfaces of constant mean curvature 206

8.6 Gaussian curvature of compact surfaces 212

9 Geodesics

9.1 Definition and basic properties 215

9.2 Geodesic equations 220

9.3 Geodesics on surfaces of revolution 227

9.4 Geodesics as shortest paths 235

9.5 Geodesic coordinates 242

10 Gauss' Theorema Egregium

10.1 The Gauss and Codazzi-Mainardi equations 247

10.2 Gauss' remarkable theorem 252

10.3 Surfaces of constant Gaussian curvature 257

10.4 Geodesic mappings 263

11 Hyperbolic geometry

11.1 Upper half-plane model 270

11.2 Isometries of H 277

11.3 Poincaré disc model 283

11.4 Hyperbolic parallels 290

11.5 Beltrami-Klein model 295

12 Minimal surfaces

12.1 Plateau's problem 305

12.2 Examples of minimal surfaces 312

12.3 Gauss map of a minimal surface 320

12.4 Conformal parametrization of minimal surfaces 322

12.5 Minimal surfaces and holomorphic functions 325

13 The Gauss-Bonnet theorem

13.1 Gauss-Bonnet for simple closed curves 335

13.2 Gauss-Bonnet for curvilinear polygons 342

13.3 Integration on compact surfaces 346

13.4 Gauss-Bonnet for compact surfaces 349

13.5 Map colouring 357

13.6 Holonomy and Gaussian curvature 362

13.7 Singularities of vector fields 365

13.8 Critical points 372

A0 Inner product spaces and self-adjoint linear maps

Al Isometries of Euclidean spaces

A2 Möbius transformations

Hints to selected exercises



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