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# Elementary Differential Geometry

**by**Pressley, Andrew

2nd

### 9781848828902

184882890X

Paperback

2/14/2010

Springer Verlag

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## Customer Reviews

Good Introduction August 17, 2011

by

by

After trying several others, I found this textbook the best for individual study. The worked out solutions to all exercises gives you a good way to check your understanding. The writing is clear and the textbook has adequate illustrations to help you see what's going on. I also liked presentation in Banchoff's book, Differential Geometry of Curves and Surfaces, but it has no answers or solutions to the exercises.

Elementary Differential Geometry:
stars based on
1 user reviews.

## Summary

The Springer Undergraduate Mathematics Series (SUMS) is designed for undergraduates in the mathematical sciences. From core foundational material to final year topics, SUMS books take a fresh and modern approach and are ideal for self-study or for a one- or two-semester course. Each book includes numerous examples, problems and fully-worked solutions.

## Table of Contents

Preface | |

Contents | |

Curves in the plane and in space | |

What is a curve? | p. 1 |

Arc-length | p. 9 |

Reparametrization | p. 13 |

Closed curves | p. 19 |

Level curves versus parametrized curves | p. 23 |

How much does a curve curve? | |

Curvature | p. 29 |

Plane curves | p. 34 |

Space curves | p. 46 |

Global properties of curves | |

Simple closed curves | p. 55 |

The isoperimetric inequality | p. 58 |

The four vertex theorem | p. 62 |

Surfaces in three dimensions | |

What is a surface? | p. 67 |

Smooth surfaces | p. 76 |

Smooth maps | p. 82 |

Tangents and derivatives | p. 85 |

Normals and orientability | p. 89 |

Examples of surfaces | |

Level surfaces | p. 95 |

Quadric surfaces | p. 97 |

Ruled surfaces and surfaces of revolution | p. 104 |

Compact surfaces | p. 109 |

Triply orthogonal systems | p. 111 |

Applications of the inverse function theorem | p. 116 |

The first fundamental form | |

Lengths of curves on surfaces | p. 121 |

Isometries of surfaces | p. 126 |

Conformal mappings of surfaces | p. 133 |

Equiareal maps mid a theorem of Archimedes | p. 139 |

Spherical geometry | p. 148 |

Curvature of surfaces | |

The second fundamental form | p. 159 |

The Gauss and Weingarten maps | p. 162 |

Normal and geodesic curvatures | p. 165 |

Parallel transport and covariant derivative | p. 170 |

Gaussian, mean and principal curvatures | |

Gaussian and mean curvatures | p. 179 |

Principal curvatures of a surface | p. 187 |

Surfaces of constant Gaussian curvature | p. 196 |

Flat surfaces | p. 201 |

Surfaces of constant mean curvature | p. 206 |

Gaussian curvature of compact surfaces | p. 212 |

Geodesics | |

Definition and basic properties | p. 215 |

Geodesic equations | p. 220 |

Geodesics on surfaces of revolution | p. 227 |

Geodesics as shortest paths | p. 235 |

Geodesic coordinates | p. 242 |

Gauss' Theorema Egregium | |

The Gauss and Codazzi-Mainardi equations | p. 247 |

Gauss' remarkable theorem | p. 252 |

Surfaces of constant Gaussian curvature | p. 257 |

Geodesic mappings | p. 263 |

Hyperbolic geometry | |

Upper half-plane model | p. 270 |

Isometries of H | p. 277 |

Poincaré disc model | p. 283 |

Hyperbolic parallels | p. 290 |

Beltrami-Klein model | p. 295 |

Minimal surfaces | |

Plateau's problem | p. 305 |

Examples of minimal surfaces | p. 312 |

Gauss map of a minimal surface | p. 320 |

Conformal parametrization of minimal surfaces | p. 322 |

Minimal surfaces and holomorphic functions | p. 325 |

The Gauss-Bonnet theorem | |

Gauss-Bonnet for simple closed curves | p. 335 |

Gauss-Bonnet for curvilinear polygons | p. 342 |

Integration on compact surfaces | p. 346 |

Gauss-Bonnet for compact surfaces | p. 349 |

Map colouring | p. 357 |

Holonomy and Gaussian curvature | p. 362 |

Singularities of vector fields | p. 365 |

Critical points | p. 372 |

Inner product spaces and self-adjoint linear maps | |

Isometries of Euclidean spaces | |

Möbius transformations | |

Hints to selected exercises | |

Solutions | |

Index | |

Table of Contents provided by Ingram. All Rights Reserved. |