Overview
- Illuminates the mathematical theory behind modern geometry processing
- Offers a uniquely accessible entry-point that is suitable for students and professionals alike
- Builds the mathematical theory behind modern applications in medical imaging, computer vision, robotics, and machine learning
- Includes exercises throughout that are suitable for class use or independent study
Part of the book series: Geometry and Computing (GC, volume 12)
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Table of contents (23 chapters)
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Front Matter
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Introduction to Differential Manifolds and Lie Groups
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Front Matter
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Riemannian Geometry, Lie Groups, and Homogeneous Spaces
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Front Matter
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Keywords
- Differential geometry for computing
- differential geometry for geometry processing
- differential geometry textbook
- differential geometry for computer vision
- differential geometry for robotics
- differential geometry for machine learning
- homogeneous spaces
- matrix lie groups
- matrix exponential
- adjoint representation
- linear lie groups
- grassmannian manifold
- stiefel manifold
- lie algebras for computing
- lie brackets
- Lorentz groups
- Riemannian manifold
- Riemannian manifold curvature
- Connections on real manifolds
- Theory of manifold optimization techniques
About this book
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.
Reviews
“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)
Authors and Affiliations
About the authors
Jocelyn Quaintance is postdoctoral researcher at the University of Pennsylvania who has contributed to the fields of combinatorial identities and power product expansions. Her recent mathematical books investigate the interplay between mathematics and computer science. Covering areas as diverse as differential geometry, linear algebra, optimization theory, and Fourier analysis, her writing illuminates the mathematics behind topics relevant to engineering, computer vision, and robotics.
Bibliographic Information
Book Title: Differential Geometry and Lie Groups
Book Subtitle: A Computational Perspective
Authors: Jean Gallier, Jocelyn Quaintance
Series Title: Geometry and Computing
DOI: https://doi.org/10.1007/978-3-030-46040-2
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Switzerland AG 2020
Hardcover ISBN: 978-3-030-46039-6Published: 15 August 2020
Softcover ISBN: 978-3-030-46042-6Published: 16 August 2021
eBook ISBN: 978-3-030-46040-2Published: 14 August 2020
Series ISSN: 1866-6795
Series E-ISSN: 1866-6809
Edition Number: 1
Number of Pages: XV, 777
Number of Illustrations: 1 b/w illustrations, 32 illustrations in colour
Topics: Differential Geometry, Topological Groups, Lie Groups, Computational Mathematics and Numerical Analysis