**Graduate Studies in Mathematics**

Volume: 178;
2017;
414 pp;
Hardcover

MSC: Primary 22; 53; 58;

Print ISBN: 978-1-4704-2952-2

Product Code: GSM/178

List Price: $73.00

Individual Member Price: $58.40

**Electronic ISBN: 978-1-4704-3747-3
Product Code: GSM/178.E**

List Price: $73.00

Individual Member Price: $58.40

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#### Supplemental Materials

# From Frenet to Cartan: The Method of Moving Frames

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*Jeanne N. Clelland*

The method of moving frames originated in the early
nineteenth century with the notion of the Frenet frame along
a curve in Euclidean space. Later, Darboux expanded this
idea to the study of surfaces. The method was brought to
its full power in the early twentieth century by Elie
Cartan, and its development continues today with the work of
Fels, Olver, and others.

This book is an introduction to the method of moving frames as
developed by Cartan, at a level suitable for beginning graduate
students familiar with the geometry of curves and surfaces in
Euclidean space. The main focus is on the use of this method to
compute local geometric invariants for curves and surfaces in various
3-dimensional homogeneous spaces, including Euclidean, Minkowski,
equi-affine, and projective spaces. Later chapters include
applications to several classical problems in differential geometry,
as well as an introduction to the nonhomogeneous case via moving
frames on Riemannian manifolds.

The book is written in a reader-friendly style, building on already
familiar concepts from curves and surfaces in Euclidean space. A
special feature of this book is the inclusion of detailed guidance
regarding the use of the computer algebra system Maple™ to
perform many of the computations involved in the exercises.

An excellent and unique graduate level exposition of the differential geometry of curves, surfaces and higher-dimensional submanifolds of homogeneous spaces based on the powerful and elegant method of moving frames. The treatment is self-contained and illustrated through a large number of examples and exercises, augmented by Maple code to assist in both concrete calculations and plotting. Highly recommended.

—Niky Kamran, McGill University

The method of moving frames has seen a tremendous explosion of research activity in recent years, expanding into many new areas of applications, from computer vision to the calculus of variations to geometric partial differential equations to geometric numerical integration schemes to classical invariant theory to integrable systems to infinite-dimensional Lie pseudo-groups and beyond. Cartan theory remains a touchstone in modern differential geometry, and Clelland's book provides a fine new introduction that includes both classic and contemporary geometric developments and is supplemented by Maple symbolic software routines that enable the reader to both tackle the exercises and delve further into this fascinating and important field of contemporary mathematics.

Recommended for students and researchers wishing to expand their geometric horizons.

—Peter Olver, University of Minnesota

#### Readership

Undergraduate and graduate students interested in differential geometry.

#### Reviews & Endorsements

This volume provides a well-written and accessible introduction to Cartan's theory of moving frames for curves and surfaces in several 3-dimensional geometries.

-- Francis Valiquette, Mathematical Reviews

Primarily intended for 'beginning graduate students,' this book is highly recommended to anyone seeking to extend their knowledge of differential geometry beyond the undergraduate level.

-- Peter Ruane, MAA Reviews

#### Table of Contents

# Table of Contents

## From Frenet to Cartan: The Method of Moving Frames

- Cover Cover11
- Title page iii4
- Contents vii8
- Preface xi12
- Acknowledgments xv16
- Part 1 . Background material 118
- Chapter 1. Assorted notions from differential geometry 320
- Chapter 2. Differential forms 3552
- 2.1. Introduction 3552
- 2.2. Dual spaces, the cotangent bundle, and tensor products 3552
- 2.3. 1-forms on \bb{𝑅}ⁿ 4057
- 2.4. 𝑝-forms on \bb{𝑅}ⁿ 4158
- 2.5. The exterior derivative 4360
- 2.6. Closed and exact forms and the Poincaré lemma 4663
- 2.7. Differential forms on manifolds 4764
- 2.8. Pullbacks 4966
- 2.9. Integration and Stokes’s theorem 5370
- 2.10. Cartan’s lemma 5572
- 2.11. The Lie derivative 5673
- 2.12. Introduction to the Cartan package for Maple 5976

- Part 2 . Curves and surfaces in homogeneous spaces via the method of moving frames 6784
- Chapter 3. Homogeneous spaces 6986
- Chapter 4. Curves and surfaces in Euclidean space 107124
- 4.1. Introduction 107124
- 4.2. Equivalence of submanifolds of a homogeneous space 108125
- 4.3. Moving frames for curves in \bb{𝐸}³ 111128
- 4.4. Compatibility conditions and existence of submanifolds with prescribed invariants 115132
- 4.5. Moving frames for surfaces in \bb{𝐸}³ 117134
- 4.6. Maple computations 134151

- Chapter 5. Curves and surfaces in Minkowski space 143160
- Chapter 6. Curves and surfaces in equi-affine space 171188
- Chapter 7. Curves and surfaces in projective space 203220

- Part 3 . Applications of moving frames 249266
- Part 4 . Beyond the flat case: Moving frames on Riemannian manifolds 337354
- Chapter 11. Curves and surfaces in elliptic and hyperbolic spaces 339356
- Chapter 12. The nonhomogeneous case: Moving frames on Riemannian manifolds 361378
- 12.1. Introduction 361378
- 12.2. Orthonormal frames and connections on Riemannian manifolds 362379
- 12.3. The Levi-Civita connection 370387
- 12.4. The structure equations 373390
- 12.5. Moving frames for curves in 3-dimensional Riemannian manifolds 379396
- 12.6. Moving frames for surfaces in 3-dimensional Riemannian manifolds 381398
- 12.7. Maple computations 388405

- Bibliography 397414
- Index 403420

- Back Cover Back Cover1433