Overview
- Explores the advanced mathematical theory behind modern geometry processing
- Offers a uniquely accessible approach that is suitable for students and professionals alike
- Augments core topics in advanced differential geometry with analytic and algebraic perspectives
- Includes exercises throughout that are suitable for class use or independent study
Part of the book series: Geometry and Computing (GC, volume 13)
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Table of contents (12 chapters)
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Front Matter
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Back Matter
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
- Tensor algebra
- Differential forms
- Frobenius theorem
- Tensor products
- Stokes' theorem
- Clifford algebras
- Clifford groups
- Spherical harmonics
- Riemannian manifolds
- Pinor and Spinor groups
- Pin group
- Spin group
- Vector bundles
- Curvature form
About this book
This textbook explores advanced topics in differential geometry, chosen for their particular relevance to modern geometry processing. Analytic and algebraic perspectives augment core topics, with the authors taking care to motivate each new concept. Whether working toward theoretical or applied questions, readers will appreciate this accessible exploration of the mathematical concepts behind many modern applications.
Beginning with an in-depth study of tensors and differential forms, the authors go on to explore a selection of topics that showcase these tools. An analytic theme unites the early chapters, which cover distributions, integration on manifolds and Lie groups, spherical harmonics, and operators on Riemannian manifolds. An exploration of bundles follows, from definitions to connections and curvature in vector bundles, culminating in a glimpse of Pontrjagin and Chern classes. The final chapter on Clifford algebras and Clifford groups draws the book to an algebraic conclusion, which can be seen as a generalized viewpoint of the quaternions.
Differential Geometry and Lie Groups: A Second Course captures the mathematical theory needed for advanced study in differential geometry with a view to furthering geometry processing capabilities. Suited to classroom use or independent study, the text will appeal to students and professionals alike. A first course in differential geometry is assumed; the authors’ companion volume Differential Geometry and Lie Groups: A Computational Perspective provides the ideal preparation.
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 Second Course
Authors: Jean Gallier, Jocelyn Quaintance
Series Title: Geometry and Computing
DOI: https://doi.org/10.1007/978-3-030-46047-1
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Switzerland AG 2020
Hardcover ISBN: 978-3-030-46046-4Published: 19 August 2020
Softcover ISBN: 978-3-030-46049-5Published: 19 August 2021
eBook ISBN: 978-3-030-46047-1Published: 18 August 2020
Series ISSN: 1866-6795
Series E-ISSN: 1866-6809
Edition Number: 1
Number of Pages: XIV, 620
Number of Illustrations: 78 b/w illustrations, 32 illustrations in colour
Topics: Differential Geometry, Topological Groups, Lie Groups, Computational Mathematics and Numerical Analysis