LEADER 05315nam 22006255 450 001 996418267903316 005 20200814030709.0 010 $a3-030-46040-1 024 7 $a10.1007/978-3-030-46040-2 035 $a(CKB)4100000011384244 035 $a(DE-He213)978-3-030-46040-2 035 $a(MiAaPQ)EBC6301520 035 $a(Au-PeEL)EBL6301520 035 $a(OCoLC)1195712242 035 $a(PPN)269146911 035 $a(EXLCZ)994100000011384244 100 $a20200814d2020 u| 0 101 0 $aeng 135 $aurnn#008mamaa 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aDifferential Geometry and Lie Groups$b[electronic resource] $eA Computational Perspective /$fby Jean Gallier, Jocelyn Quaintance 205 $a1st ed. 2020. 210 1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2020. 215 $a1 online resource (XV, 777 p. 33 illus., 32 illus. in color.) 225 1 $aGeometry and Computing,$x1866-6795 ;$v12 311 $a3-030-46039-8 327 $a1. 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. 330 $aThis textbook offers an introduction to differential geometry designed for readers interested in modern geometry processing. Working from basic undergraduate prerequisites, the authors develop manifold theory and Lie groups from scratch; fundamental topics in Riemannian geometry follow, culminating in the theory that underpins manifold optimization techniques. Students and professionals working in computer vision, robotics, and machine learning will appreciate this pathway into the mathematical concepts behind many modern applications. Starting with the matrix exponential, the text begins with an introduction to Lie groups and group actions. Manifolds, tangent spaces, and cotangent spaces follow; a chapter on the construction of manifolds from gluing data is particularly relevant to the reconstruction of surfaces from 3D meshes. Vector fields and basic point-set topology bridge into the second part of the book, which focuses on Riemannian geometry. Chapters on Riemannian manifolds encompass Riemannian metrics, geodesics, and curvature. Topics that follow include submersions, curvature on Lie groups, and the Log-Euclidean framework. The final chapter highlights naturally reductive homogeneous manifolds and symmetric spaces, revealing the machinery needed to generalize important optimization techniques to Riemannian manifolds. Exercises are included throughout, along with optional sections that delve into more theoretical topics. Differential Geometry and Lie Groups: A Computational Perspective offers a uniquely accessible perspective on differential geometry for those interested in the theory behind modern computing applications. Equally suited to classroom use or independent study, the text will appeal to students and professionals alike; only a background in calculus and linear algebra is assumed. Readers looking to continue on to more advanced topics will appreciate the authors? companion volume Differential Geometry and Lie Groups: A Second Course. 410 0$aGeometry and Computing,$x1866-6795 ;$v12 606 $aDifferential geometry 606 $aTopological groups 606 $aLie groups 606 $aComputer mathematics 606 $aDifferential Geometry$3https://scigraph.springernature.com/ontologies/product-market-codes/M21022 606 $aTopological Groups, Lie Groups$3https://scigraph.springernature.com/ontologies/product-market-codes/M11132 606 $aComputational Mathematics and Numerical Analysis$3https://scigraph.springernature.com/ontologies/product-market-codes/M1400X 615 0$aDifferential geometry. 615 0$aTopological groups. 615 0$aLie groups. 615 0$aComputer mathematics. 615 14$aDifferential Geometry. 615 24$aTopological Groups, Lie Groups. 615 24$aComputational Mathematics and Numerical Analysis. 676 $a516.36 700 $aGallier$b Jean$4aut$4http://id.loc.gov/vocabulary/relators/aut$0510781 702 $aQuaintance$b Jocelyn$4aut$4http://id.loc.gov/vocabulary/relators/aut 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a996418267903316 996 $aDifferential Geometry and Lie Groups$91886251 997 $aUNISA