From Frenet to Cartan: The Method of Moving Frames (Graduate Studies in Mathematics) (Graduate Studies in Mathematics, 178)

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 Cover Title page Contents Preface Acknowledgments Part 1 . Background material Chapter 1. Assorted notions from differential geometry 1.1. Manifolds 1.2. Tensors, indices, and the Einstein summation convention 1.3. Differentiable maps, tangent spaces, and vector fields 1.4. Lie groups and matrix groups 1.5. Vector bundles and principal bundles Chapter 2. Differential forms 2.1. Introduction 2.2. Dual spaces, the cotangent bundle, and tensor products 2.3. 1-forms on \bb{R}n 2.4. p-forms on \bb{R}n 2.5. The exterior derivative 2.6. Closed and exact forms and the Poincaré lemma 2.7. Differential forms on manifolds 2.8. Pullbacks 2.9. Integration and Stokes’s theorem 2.10. Cartan’s lemma 2.11. The Lie derivative 2.12. Introduction to the Cartan package for Maple Part 2 . Curves and surfaces in homogeneous spaces via the method of moving frames Chapter 3. Homogeneous spaces 3.1. Introduction 3.2. Euclidean space 3.3. Orthonormal frames on Euclidean space 3.4. Homogeneous spaces 3.5. Minkowski space 3.6. Equi-affine space 3.7. Projective space 3.8. Maple computations Chapter 4. Curves and surfaces in Euclidean space 4.1. Introduction 4.2. Equivalence of submanifolds of a homogeneous space 4.3. Moving frames for curves in \bb{E}3 4.4. Compatibility conditions and existence of submanifolds with prescribed invariants 4.5. Moving frames for surfaces in \bb{E}3 4.6. Maple computations Chapter 5. Curves and surfaces in Minkowski space 5.1. Introduction 5.2. Moving frames for timelike curves in \M^{1,2} 5.3. Moving frames for timelike surfaces in \M^{1,2} 5.4. An alternate construction for timelike surfaces 5.5. Maple computations Chapter 6. Curves and surfaces in equi-affine space 6.1. Introduction 6.2. Moving frames for curves in \bb{A}3 6.3. Moving frames for surfaces in \bb{A}3 6.4. Maple computations Chapter 7. Curves and surfaces in projective space 7.1. Introduction 7.2. Moving frames for curves in \bb{P}2 7.3. Moving frames for curves in \bb{P}3 7.4. Moving frames for surfaces in \bb{P}3 7.5. Maple computations Part 3 . Applications of moving frames Chapter 8. Minimal surfaces in \E3 and \bb{A}3 8.1. Introduction 8.2. Minimal surfaces in \bb{E}3 8.3. Minimal surfaces in \bb{A}3 8.4. Maple computations Chapter 9. Pseudospherical surfaces and Bäcklund’s theorem 9.1. Introduction 9.2. Line congruences 9.3. Bäcklund’s theorem 9.4. Pseudospherical surfaces and the sine-Gordon equation 9.5. The Bäcklund transformation for the sine-Gordon equation 9.6. Maple computations Chapter 10. Two classical theorems 10.1. Doubly ruled surfaces in \R3 10.2. The Cauchy-Crofton formula 10.3. Maple computations Part 4 . Beyond the flat case: Moving frames on Riemannian manifolds Chapter 11. Curves and surfaces in elliptic and hyperbolic spaces 11.1. Introduction 11.2. The homogeneous spaces \bb{S}n and \bb{H}n 11.3. A more intrinsic view of \bb{S}n and \bb{H}n 11.4. Moving frames for curves in \bb{S}3 and \bb{H}3 11.5. Moving frames for surfaces in \bb{S}3 and \bb{H}3 11.6. Maple computations Chapter 12. The nonhomogeneous case: Moving frames on Riemannian manifolds 12.1. Introduction 12.2. Orthonormal frames and connections on Riemannian manifolds 12.3. The Levi-Civita connection 12.4. The structure equations 12.5. Moving frames for curves in 3-dimensional Riemannian manifolds 12.6. Moving frames for surfaces in 3-dimensional Riemannian manifolds 12.7. Maple computations Bibliography Index Back Cover 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 MapleTM 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 "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 MapleTM to perform many of the computations involved in the exercises"--amazon.com