Differential Geometry and Lie Groups, 1st ed. 2020 A Computational Perspective Geometry and Computing Series, Vol. 12
Auteurs : Gallier Jean, Quaintance Jocelyn
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.
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.
Date de parution : 08-2021
Ouvrage de 777 p.
15.5x23.5 cm
Date de parution : 08-2020
Ouvrage de 777 p.
15.5x23.5 cm
Mots-clés :
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