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Differential Geometry and Lie Groups

A Computational Perspective (Sprache: Englisch)
Autoren: Jean Gallier , Jocelyn Quaintance
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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...
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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...
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Produktdetails
Produktinformationen zu „Differential Geometry and Lie Groups “
Klappentext zu „Differential Geometry and Lie Groups “
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 .
Inhaltsverzeichnis zu „Differential Geometry and Lie Groups “
1. The Matrix Exponential; Some Matrix Lie Groups.- 2. Adjoint Representations and the Derivative of exp.- 3. Introduction to Manifolds and Lie Groups.- 4. Groups and Group Actions.- 5. The Lorentz Groups .- 6. The Structure of O( p , q ) and SO( p , q ).- 7. Manifolds, Tangent Spaces, Cotangent Spaces.- 8. Construction of Manifolds From Gluing Data .- 9. Vector Fields, Integral Curves, Flows.- 10. Partitions of Unity, Covering Maps .- 11. Basic Analysis: Review of Series and Derivatives.- 12. A Review of Point Set Topology.-13. Riemannian Metrics, Riemannian Manifolds.- 14. Connections on Manifolds.- 15. Geodesics on Riemannian Manifolds.- 16. Curvature in Riemannian Manifolds.- 17. Isometries, Submersions, Killing Vector Fields.- 18. Lie Groups, Lie Algebra, Exponential Map.- 19. The Derivative of exp and Dynkin's Formula .- 20. Metrics, Connections, and Curvature of Lie Groups.- 21. The Log-Euclidean Framework.- 22. Manifolds Arising from Group Actions.
Autoren-Porträt von Jean Gallier, Jocelyn Quaintance
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.
Bibliographische Angaben

  • Autoren: Jean Gallier , Jocelyn Quaintance
  • 2020, 1st ed., xv, 777 Seiten, 32 farbige Abbildungen, 1 Schwarz-Weiß-Abbildungen, Maße: 16,4 x 24,4 cm, Gebunden, Englisch
  • Verlag: Springer, Berlin
  • ISBN-10: 3030460398
  • ISBN-13: 9783030460396
  • Erscheinungsdatum: 15.08.2020

Sprache:
Englisch
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