2017;
105 pp;
Softcover

MSC: Primary 51; 53;

Print ISBN: 978-1-4704-3716-9

Product Code: MBK/110

List Price: $49.00

AMS Member Price: $39.20

**Electronic ISBN: 978-1-4704-4307-8
Product Code: MBK/110.E**

List Price: $49.00

AMS Member Price: $39.20

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#### Supplemental Materials

# Non-Euclidean Geometry and Curvature: Two-Dimensional Spaces, Volume 3

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*James W. Cannon*

This is the final volume of a three volume
collection devoted to the geometry, topology, and curvature of
2-dimensional spaces. The collection provides a guided tour through a
wide range of topics by one of the twentieth century's masters of
geometric topology. The books are accessible to college and graduate
students and provide perspective and insight to mathematicians at all
levels who are interested in geometry and topology.

Einstein showed how to interpret gravity as the dynamic response to
the curvature of space-time. Bill Thurston showed us that
non-Euclidean geometries and curvature are essential to the
understanding of low-dimensional spaces. This third and final volume
aims to give the reader a firm intuitive understanding of these
concepts in dimension 2. The volume first demonstrates a number of the
most important properties of non-Euclidean geometry by means of simple
infinite graphs that approximate that geometry. This is followed by a
long chapter taken from lectures the author gave at MSRI, which
explains a more classical view of hyperbolic non-Euclidean geometry in
all dimensions. Finally, the author explains a natural intrinsic
obstruction to flattening a triangulated polyhedral surface into the
plane without distorting the constituent triangles. That obstruction
extends intrinsically to smooth surfaces by approximation and is
called curvature. Gauss's original definition of curvature is
extrinsic rather than intrinsic. The final two chapters show that the
book's intrinsic definition is equivalent to Gauss's extrinsic
definition (Gauss's “Theorema Egregium” (“Great
Theorem”)).

#### Readership

Graduate and undergraduate students and researchers interested in topology.

#### Table of Contents

# Table of Contents

## Non-Euclidean Geometry and Curvature: Two-Dimensional Spaces, Volume 3

- Cover Cover11
- Title page iii4
- Contents v6
- Preface to the Three Volume Set vii8
- Preface to Volume 3 xi12
- Chapter 1. A Graphical Introduction to Hyperbolic Geometry 114
- Chapter 2. Hyperbolic Geometry 1124
- 2.1. Introduction 1124
- 2.2. The Origins of Hyperbolic Geometry 1225
- 2.3. Why Call It Hyperbolic Geometry? 1427
- 2.4. Understanding the One-dimensional Case 1528
- 2.5. Generalizing to Higher Dimensions 1831
- 2.6. Rudiments of Riemannian Geometry 1831
- 2.7. Five Models of Hyperbolic Space 1932
- 2.8. Stereographic Projection 2235
- 2.9. Geodesics 2639
- 2.10. Isometries and Distances in the Hyperboloid Model 3043
- 2.11. The Space at Infinity 3245
- 2.12. The Geometric Classification of Isometries 3346
- 2.13. Curious Facts about Hyperbolic Space 3447
- 2.14. The Sixth Model 4356
- 2.15. Why Study Hyperbolic Geometry? 4558
- 2.16. When Does a Manifold Have a Hyperbolic Structure? 4962
- 2.17. How to get Analytic Coordinates at Infinity? 5265

- Chapter 3. Gravity As Curvature 5568
- Chapter 4. Curvature by Polyhedral Approximation 5770
- 4.1. Approximating Smooth Surfaces by Polyhedra 5770
- 4.2. The Curvature of a Polyhedral Disk 5770
- 4.3. How Flat Is a Disk? 5770
- 4.4. How Straight Is a Disk Boundary? 5972
- 4.5. Duality Theorem: Angle Defect + Boundary Defect = 2𝜋 6174
- 4.6. The Curvature of a Polyhedral Disk 6376
- 4.7. Applications of the Duality Between Angle Defect and Boundary Defect 6376
- 4.8. The Curvature of a Smooth Disk 6679

- Chapter 5. Curvature As a Length Derivative 6982
- Chapter 6. Theorema Egregium 8194
- Chapter 7. Curvature Appendix 87100
- 7.1. The Generalized Umlauf Theorem 87100
- 7.2. Two Technical Properties of a Smooth Surface 88101
- 7.3. Specialized Polyhedral Approximations to 𝐷’. 90103
- 7.4. Specialized Polyhedral Approximations in the Plane. 90103
- 7.5. Specialized Polyhedral Approximations on Curved Surfaces. 92105
- 7.6. Outline of the Proof of the Theorem. 94107
- 7.7. Exercises 98111

- Bibliography 99112
- Back Cover Back Cover1119