Hardcover ISBN:  9781470429522 
Product Code:  GSM/178 
List Price:  $135.00 
MAA Member Price:  $121.50 
AMS Member Price:  $108.00 
eBook ISBN:  9781470437473 
Product Code:  GSM/178.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Hardcover ISBN:  9781470429522 
eBook: ISBN:  9781470437473 
Product Code:  GSM/178.B 
List Price:  $220.00 $177.50 
MAA Member Price:  $198.00 $159.75 
AMS Member Price:  $176.00 $142.00 
Hardcover ISBN:  9781470429522 
Product Code:  GSM/178 
List Price:  $135.00 
MAA Member Price:  $121.50 
AMS Member Price:  $108.00 
eBook ISBN:  9781470437473 
Product Code:  GSM/178.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Hardcover ISBN:  9781470429522 
eBook ISBN:  9781470437473 
Product Code:  GSM/178.B 
List Price:  $220.00 $177.50 
MAA Member Price:  $198.00 $159.75 
AMS Member Price:  $176.00 $142.00 

Book DetailsGraduate Studies in MathematicsVolume: 178; 2017; 414 ppMSC: Primary 22; 53; 58
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 3dimensional homogeneous spaces, including Euclidean, Minkowski, equiaffine, 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 readerfriendly 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^{TM} 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 higherdimensional submanifolds of homogeneous spaces based on the powerful and elegant method of moving frames. The treatment is selfcontained 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 infinitedimensional Lie pseudogroups 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
ReadershipUndergraduate and graduate students interested in differential geometry.

Table of Contents

Background material

Assorted notions from differential geometry

Differential forms

Curves and surfaces in homogeneous spaces via the method of moving frames

Homogeneous spaces

Curves and surfaces in Euclidean space

Curves and surfaces in Minkowski space

Curves and surfaces in equiaffine space

Curves and surfaces in projective space

Applications of moving frames

Minimal surfaces in $\mathbb {E}^3$ and $\mathbb {A}^3$

Pseudospherical surfaces in Bäcklund’s theorem

Two classical theorems

Beyond the flat case: Moving frames on Riemannian manifolds

Curves and surfaces in elliptic and hyperbolic spaces

The nonhomogeneous case: Moving frames on Riemannian manifolds


Additional Material

Reviews

The present book has a high didactical and scientific quality being a very careful introduction to the method of moving frames...this book is a very nice presentation of an essential tool of classical differential geometry. I strongly recommend it as a welcome addition to the main textbooks in geometry.
Mircea Crâşmăreanu, Zentralblatt MATH 
This volume provides a wellwritten and accessible introduction to Cartan's theory of moving frames for curves and surfaces in several 3dimensional 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


RequestsReview Copy – for publishers of book reviewsDesk Copy – for instructors who have adopted an AMS textbook for a courseExamination Copy – for faculty considering an AMS textbook for a coursePermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Additional Material
 Reviews
 Requests
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 3dimensional homogeneous spaces, including Euclidean, Minkowski, equiaffine, 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 readerfriendly 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^{TM} 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 higherdimensional submanifolds of homogeneous spaces based on the powerful and elegant method of moving frames. The treatment is selfcontained 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 infinitedimensional Lie pseudogroups 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
Undergraduate and graduate students interested in differential geometry.

Background material

Assorted notions from differential geometry

Differential forms

Curves and surfaces in homogeneous spaces via the method of moving frames

Homogeneous spaces

Curves and surfaces in Euclidean space

Curves and surfaces in Minkowski space

Curves and surfaces in equiaffine space

Curves and surfaces in projective space

Applications of moving frames

Minimal surfaces in $\mathbb {E}^3$ and $\mathbb {A}^3$

Pseudospherical surfaces in Bäcklund’s theorem

Two classical theorems

Beyond the flat case: Moving frames on Riemannian manifolds

Curves and surfaces in elliptic and hyperbolic spaces

The nonhomogeneous case: Moving frames on Riemannian manifolds

The present book has a high didactical and scientific quality being a very careful introduction to the method of moving frames...this book is a very nice presentation of an essential tool of classical differential geometry. I strongly recommend it as a welcome addition to the main textbooks in geometry.
Mircea Crâşmăreanu, Zentralblatt MATH 
This volume provides a wellwritten and accessible introduction to Cartan's theory of moving frames for curves and surfaces in several 3dimensional 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