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  • Cited by 8
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    • Publisher:
      Cambridge University Press
      Publication date:
      December 2022
      December 2022
      ISBN:
      9781009091251
      9781316514887
      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
      Dimensions:
      (235 x 158 mm)
      Weight & Pages:
      0.57kg, 282 Pages
      Dimensions:
      Weight & Pages:
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    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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