An Introduction to Infinite-Dimensional Differential Geometry

Alexander Schmeding

doi:10.1017/9781009091251

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Grong, Erlend Nilssen, Torstein and Schmeding, Alexander 2022. Geometric rough paths on infinite dimensional spaces. Journal of Differential Equations, Vol. 340, Issue. , p. 151.

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Gieres, Francois 2023. Covariant canonical formulations of classical field theories. SciPost Physics Lecture Notes,

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Oertel, Frank 2024. Upper Bounds for Grothendieck Constants, Quantum Correlation Matrices and CCP Functions. Vol. 2349, Issue. , p. 39.

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Grong, Erlend and Schmeding, Alexander 2024. Controllability and diffeomorphism groups on manifolds with boundary. Forum Mathematicum,

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Lucas, Javier de Lange, Julia and Rivas, Xavier 2024. A symplectic approach to Schrödinger equations in the infinite-dimensional unbounded setting. AIMS Mathematics, Vol. 9, Issue. 10, p. 27998.

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Alexander Schmeding, Nord Universitet, Norway

Publisher:: Cambridge University Press
Online publication date:: December 2022
Print publication year:: 2022
Online ISBN:: 9781009091251
DOI:: https://doi.org/10.1017/9781009091251
Creative Commons:: 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

Subjects:: Mathematics (general), Mathematics, Mathematical Physics, Geometry and Topology
Series:: Cambridge Studies in Advanced Mathematics (202)

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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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An Introduction to Infinite-Dimensional Differential Geometry

This Book has been cited by the following publications. This list is generated based on data provided by Crossref.

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Contents

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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