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Publisher:
Cambridge University Press
Online publication date:
December 2022
Print publication year:
2022
Online ISBN:
9781009091251
Creative Commons:
Creative Common License - CC Creative Common License - BY Creative Common License - NC Creative Common License - ND
This content is Open Access and distributed under the terms of the Creative Commons Attribution licence CC-BY-NC-ND 4.0 https://creativecommons.org/creativelicenses

Book description

Introducing foundational concepts in infinite-dimensional differential geometry beyond Banach manifolds, this text is based on Bastiani calculus. It focuses on two main areas of infinite-dimensional geometry: infinite-dimensional Lie groups and weak Riemannian geometry, exploring their connections to manifolds of (smooth) mappings. Topics covered include diffeomorphism groups, loop groups and Riemannian metrics for shape analysis. Numerous examples highlight both surprising connections between finite- and infinite-dimensional geometry, and challenges occurring solely in infinite dimensions. The geometric techniques developed are then showcased in modern applications of geometry such as geometric hydrodynamics, higher geometry in the guise of Lie groupoids, and rough path theory. With plentiful exercises, some with solutions, and worked examples, this will be indispensable for graduate students and researchers working at the intersection of functional analysis, non-linear differential equations and differential geometry. This title is also available as Open Access on Cambridge Core.

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Contents

Full book PDF
  • Frontmatter
    pp i-iv
  • Contents
    pp v-vi
  • Preface
    pp vii-xiv
  • 1 - Calculus in Locally Convex Spaces
    pp 1-29
  • 2 - Spaces and Manifolds of Smooth Maps
    pp 30-47
  • 3 - Lifting Geometry to Mapping Spaces I: Lie Groups
    pp 48-79
  • 4 - Lifting Geometry to Mapping Spaces II: (Weak) Riemannian Metrics
    pp 80-105
  • 5 - Weak Riemannian Metrics with Applications in Shape Analysis
    pp 106-119
  • 6 - Connecting Finite-Dimensional, Infinite-Dimensional and Higher Geometry
    pp 120-137
  • 7 - Euler–Arnold Theory: PDEs via Geometry
    pp 138-156
  • 8 - The Geometry of Rough Paths
    pp 157-185
  • Appendix A - A Primer on Topological Vector Spaces and Locally Convex Spaces
    pp 186-205
  • Appendix B - Basic Ideas from Topology
    pp 206-212
  • Appendix C - Canonical Manifold of Mappings
    pp 213-224
  • Appendix D - Vector Fields and Their Lie Bracket
    pp 225-230
  • Appendix E - Differential Forms on Infinite-Dimensional Manifolds
    pp 231-243
  • Appendix F - Solutions to Selected Exercises
    pp 244-255
  • References
    pp 256-263
  • Index
    pp 264-267

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