Differential Geometry and Lie Groups

1. Front Matter

   Pages i-xiv

   PDF

2. Introduction

   • Jean Gallier, Jocelyn Quaintance

   Pages 1-10

3. Tensor Algebras and Symmetric Algebras

   • Jean Gallier, Jocelyn Quaintance

   Pages 11-76

4. Exterior Tensor Powers and Exterior Algebras

   • Jean Gallier, Jocelyn Quaintance

   Pages 77-127

5. Differential Forms

   • Jean Gallier, Jocelyn Quaintance

   Pages 129-193

6. Distributions and the Frobenius Theorem

   • Jean Gallier, Jocelyn Quaintance

   Pages 195-209

7. Integration on Manifolds

   • Jean Gallier, Jocelyn Quaintance

   Pages 211-263

8. Spherical Harmonics and Linear Representations of Lie Groups

   • Jean Gallier, Jocelyn Quaintance

   Pages 265-360

9. Operators on Riemannian Manifolds: Hodge Laplacian, Laplace-Beltrami Laplacian, the Bochner Laplacian, and Weitzenböck Formulae

   • Jean Gallier, Jocelyn Quaintance

   Pages 361-401

10. Bundles, Metrics on Bundles, and Homogeneous Spaces

    • Jean Gallier, Jocelyn Quaintance

    Pages 403-465

11. Connections and Curvature in Vector Bundles

    • Jean Gallier, Jocelyn Quaintance

    Pages 467-530

12. Clifford Algebras, Clifford Groups, and the Groups Pin(n) and Spin(n)

    • Jean Gallier, Jocelyn Quaintance

    Pages 531-600

13. Correction to: Differential Geometry and Lie Groups

    • Jean Gallier, Jocelyn Quaintance

    Pages C1-C1

14. Back Matter

    Pages 601-620

    PDF

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.

• 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

• Department of Computer and Information Science, University of Pennsylvania, Philadelphia, USA

  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.