Differential Geometry and Lie Groups
A Computational Perspective
Authors: Gallier, Jean, Quaintance, Jocelyn
Free Preview- 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
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- About this Textbook
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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. - About the authors
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Jean Gallier is Professor of Computer and Information Science at the University of Pennsylvania, Philadelphia. His research interests include geometry and its applications, geometric modeling, and differential geometry. He is also a member of the University of Pennsylvania’s Department of Mathematics, and its Center for Human Modelling and Simulation.
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.
- Table of contents (23 chapters)
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Introduction
Pages 1-11
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The Matrix Exponential: Some Matrix Lie Groups
Pages 15-53
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Adjoint Representations and the Derivative of exp
Pages 55-64
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Introduction to Manifolds and Lie Groups
Pages 65-116
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Groups and Group Actions
Pages 117-161
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Table of contents (23 chapters)
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Bibliographic Information
- Bibliographic Information
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- Book Title
- Differential Geometry and Lie Groups
- Book Subtitle
- A Computational Perspective
- Authors
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- Jean Gallier
- Jocelyn Quaintance
- Series Title
- Geometry and Computing
- Series Volume
- 12
- Copyright
- 2020
- Publisher
- Springer International Publishing
- Copyright Holder
- Springer Nature Switzerland AG
- eBook ISBN
- 978-3-030-46040-2
- DOI
- 10.1007/978-3-030-46040-2
- Hardcover ISBN
- 978-3-030-46039-6
- Series ISSN
- 1866-6795
- Edition Number
- 1
- Number of Pages
- XV, 777
- Number of Illustrations
- 1 b/w illustrations, 32 illustrations in colour
- Topics
