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Jean H. Gallier

Differential Geometry and Lie Groups: A Computational Perspective (Geometry and Computing, 12) 1st ed. 2020 Edition

by Jean Gallier (Author), Jocelyn Quaintance (Author)

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Book 11 of 12: Geometry and Computing

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This textbook offers an introduction to differential geometry designed for readers interested in modern geometry processing. Working from basic undergraduate prerequisites, the authors develop manifold theory and Lie groups from scratch; fundamental topics in Riemannian geometry follow, culminating in the theory that underpins manifold optimization techniques. Students and professionals working in computer vision, robotics, and machine learning will appreciate this pathway into the mathematical concepts behind many modern applications.

Starting with the matrix exponential, the text begins with an introduction to Lie groups and group actions. Manifolds, tangent spaces, and cotangent spaces follow; a chapter on the construction of manifolds from gluing data is particularly relevant to the reconstruction of surfaces from 3D meshes. Vector fields and basic point-set topology bridge into the second part of the book, which focuses on Riemannian geometry.

Chapters on Riemannian manifolds encompass Riemannian metrics, geodesics, and curvature. Topics that follow include submersions, curvature on Lie groups, and the Log-Euclidean framework. The final chapter highlights naturally reductive homogeneous manifolds and symmetric spaces, revealing the machinery needed to generalize important optimization techniques to Riemannian manifolds. Exercises are included throughout, along with optional sections that delve into more theoretical topics.

Differential Geometry and Lie Groups: A Computational Perspective offers a uniquely accessible perspective on differential geometry for those interested in the theory behind modern computing applications. Equally suited to classroom use or independent study, the text will appeal to students and professionals alike; only a background in calculus and linear algebra is assumed. Readers looking to continue on to more advanced topics will appreciate the authors’ companion volume Differential Geometry and Lie Groups: A Second Course.

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Editorial Reviews

Review

“The book … is intended ‘for a wide audience ranging from upper undergraduate to advanced graduate students in mathematics, physics, and more broadly engineering students, especially in computer science.’ … The text’s coverage is extensive, its exposition clear throughout, and the color illustrations helpful. The authors are also familiar with many texts at a comparable level and have drawn on them in several places to include some of the most insightful proofs already in the literature.” (Jer-Chin Chuang, MAA Reviews, October 4, 2021)
“The book is intended for incremental study and covers both basic concepts and more advanced ones. The former are thoroughly supported with theory and examples, and the latter are backed up with extensive reading lists and references. … Thanks to its design and approach style this is a timely and much needed addition that enables interdisciplinary bridges and the discovery of new applications for differential geometry.” (Corina Mohorian, zbMATH 1453.53001, 2021)

From the Back Cover

This textbook offers an introduction to differential geometry designed for readers interested in modern geometry processing. Working from basic undergraduate prerequisites, the authors develop manifold theory and Lie groups from scratch; fundamental topics in Riemannian geometry follow, culminating in the theory that underpins manifold optimization techniques. Students and professionals working in computer vision, robotics, and machine learning will appreciate this pathway into the mathematical concepts behind many modern applications.

Starting with the matrix exponential, the text begins with an introduction to Lie groups and group actions. Manifolds, tangent spaces, and cotangent spaces follow; a chapter on the construction of manifolds from gluing data is particularly relevant to the reconstruction of surfaces from 3D meshes. Vector fields and basic point-set topology bridge into the second part of the book, which focuses on Riemannian geometry.

Chapters on Riemannian manifolds encompass Riemannian metrics, geodesics, and curvature. Topics that follow include submersions, curvature on Lie groups, and the Log-Euclidean framework. The final chapter highlights naturally reductive homogeneous manifolds and symmetric spaces, revealing the machinery needed to generalize important optimization techniques to Riemannian manifolds. Exercises are included throughout, along with optional sections that delve into more theoretical topics.

Differential Geometry and Lie Groups: A Computational Perspective offers a uniquely accessible perspective on differential geometry for those interested in the theory behind modern computing applications. Equally suited to classroom use or independent study, the text will appeal to students and professionals alike; only a background in calculus and linear algebra is assumed. Readers looking to continue on to more advanced topics will appreciate the authors’ companion volume Differential Geometry and Lie Groups: A Second Course.

Product details

Publisher ‏ : ‎ Springer; 1st ed. 2020 edition (August 15, 2020)
Language ‏ : ‎ English
Hardcover ‏ : ‎ 792 pages
ISBN-10 ‏ : ‎ 3030460398
ISBN-13 ‏ : ‎ 978-3030460396
Item Weight ‏ : ‎ 2.91 pounds
Dimensions ‏ : ‎ 6.25 x 1.75 x 9.5 inches

Best Sellers Rank: #429,863 in Books (See Top 100 in Books)
- #38 in Differential Geometry (Books)
- #39 in Group Theory (Books)
- #69 in Counting & Numeration

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Mauricio Lopez

Exposicióm bastante técnica y rigurosa

Reviewed in Spain 🇪🇸 on March 21, 2022

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El libro tiene dos partes. La primera es una exposición bien lograda de los grupos de Lie, y los manifolds más simples e intuitivos como R^n, S^1, S^2, etc. Continúa con exposiciones cada vez más técnicas y rigurosas de los grupos de Lie GL(n), O(n), SO(n), SE(n), las algebras de Lie, el mapa exponencial, el operador diferencial, el plano proyectivo, las acciones de grupo, las transformaciones de Lorentz y mucho más.

La segunda parte es mucho mas técnica y rigurosa de la geometría Riemaniana comenzando por los manifolds abstractos, campos vectoriales en mnifolds, diferencial de Lie, diferencial covariante, geodésicas, transporte paralelo, conecciones y culmina con el la introducción de grupos de Lie y algebras de Lie en manifolds abstractos como herramienta fundamental para aplicaciones en ciencia, ingeniería y ciencia de la computacion.

Si bien tiene capítulos donde cubre vacíos en topologia y tópicos de calculo vectorial, la verdad es que, al menos la segunda parte del libro, la exposicion avanza rápidamente dejando perdido al lector que no tiene los rudimentos basicos de geometria diferencial. Por ello creo que sería recomendable acompañar el libro con un texto más lento (pero no menos riguroso) como por ejemplo el de DoCarmo.

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