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 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 Cover 1 Title page 4 Contents 8 Preface 12 Acknowledgments 16 Part 1 . Background material 18 Chapter 1. Assorted notions from differential geometry 20 1.1. Manifolds 20 1.2. Tensors, indices, and the Einstein summation convention 26 1.3. Differentiable maps, tangent spaces, and vector fields 32 1.4. Lie groups and matrix groups 43 1.5. Vector bundles and principal bundles 49 Chapter 2. Differential forms 52 2.1. Introduction 52 2.2. Dual spaces, the cotangent bundle, and tensor products 52 2.3. 1-forms on \bb{R}n 57 2.4. p-forms on \bb{R}n 58 2.5. The exterior derivative 60 2.6. Closed and exact forms and the Poincaré lemma 63 2.7. Differential forms on manifolds 64 2.8. Pullbacks 66 2.9. Integration and Stokes’s theorem 70 2.10. Cartan’s lemma 72 2.11. The Lie derivative 73 2.12. Introduction to the Cartan package for Maple 76 Part 2 . Curves and surfaces in homogeneous spaces via the method of moving frames 84 Chapter 3. Homogeneous spaces 86 3.1. Introduction 86 3.2. Euclidean space 87 3.3. Orthonormal frames on Euclidean space 92 3.4. Homogeneous spaces 101 3.5. Minkowski space 102 3.6. Equi-affine space 109 3.7. Projective space 113 3.8. Maple computations 120 Chapter 4. Curves and surfaces in Euclidean space 124 4.1. Introduction 124 4.2. Equivalence of submanifolds of a homogeneous space 125 4.3. Moving frames for curves in \bb{E}3 128 4.4. Compatibility conditions and existence of submanifolds with prescribed invariants 132 4.5. Moving frames for surfaces in \bb{E}3 134 4.6. Maple computations 151 Chapter 5. Curves and surfaces in Minkowski space 160 5.1. Introduction 160 5.2. Moving frames for timelike curves in \M^{1,2} 161 5.3. Moving frames for timelike surfaces in \M^{1,2} 166 5.4. An alternate construction for timelike surfaces 178 5.5. Maple computations 183 Chapter 6. Curves and surfaces in equi-affine space 188 6.1. Introduction 188 6.2. Moving frames for curves in \bb{A}3 189 6.3. Moving frames for surfaces in \bb{A}3 195 6.4. Maple computations 208 Chapter 7. Curves and surfaces in projective space 220 7.1. Introduction 220 7.2. Moving frames for curves in \bb{P}2 221 7.3. Moving frames for curves in \bb{P}3 231 7.4. Moving frames for surfaces in \bb{P}3 237 7.5. Maple computations 252 Part 3 . Applications of moving frames 266 Chapter 8. Minimal surfaces in \E3 and \bb{A}3 268 8.1. Introduction 268 8.2. Minimal surfaces in \bb{E}3 268 8.3. Minimal surfaces in \bb{A}3 285 8.4. Maple computations 297 Chapter 9. Pseudospherical surfaces and Bäcklund’s theorem 304 9.1. Introduction 304 9.2. Line congruences 305 9.3. Bäcklund’s theorem 306 9.4. Pseudospherical surfaces and the sine-Gordon equation 310 9.5. The Bäcklund transformation for the sine-Gordon equation 314 9.6. Maple computations 320 Chapter 10. Two classical theorems 328 10.1. Doubly ruled surfaces in \R3 328 10.2. The Cauchy-Crofton formula 341 10.3. Maple computations 346 Part 4 . Beyond the flat case: Moving frames on Riemannian manifolds 354 Chapter 11. Curves and surfaces in elliptic and hyperbolic spaces 356 11.1. Introduction 356 11.2. The homogeneous spaces \bb{S}n and \bb{H}n 357 11.3. A more intrinsic view of \bb{S}n and \bb{H}n 362 11.4. Moving frames for curves in \bb{S}3 and \bb{H}3 365 11.5. Moving frames for surfaces in \bb{S}3 and \bb{H}3 368 11.6. Maple computations 374 Chapter 12. The nonhomogeneous case: Moving frames on Riemannian manifolds 378 12.1. Introduction 378 12.2. Orthonormal frames and connections on Riemannian manifolds 379 12.3. The Levi-Civita connection 387 12.4. The structure equations 390 12.5. Moving frames for curves in 3-dimensional Riemannian manifolds 396 12.6. Moving frames for surfaces in 3-dimensional Riemannian manifolds 398 12.7. Maple computations 405 Bibliography 414 Index 420 Back Cover 433 "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 Background material: Assorted notions from differential geometryDifferential formsCurves and surfaces in homogeneous spaces via the method of moving frames: Homogeneous spacesCurves and surfaces in Euclidean spaceCurves and surfaces in Minkowski spaceCurves and surfaces in equi-affine spaceCurves and surfaces in projective spaceApplications of moving frames: Minimal surfaces in $\mathbb{E}^3$ and $\mathbb{A}^3$Pseudospherical surfaces in Backlund's theoremTwo classical theoremsBeyond the flat case: Moving frames on Riemannian manifolds: Curves and surfaces in elliptic and hyperbolic spacesThe nonhomogeneous case: Moving frames on Riemannian manifoldsBibliographyIndex.