Book contents
- Frontmatter
- Contents
- Preface
- 1 Calculus in Locally Convex Spaces
- 2 Spaces and Manifolds of Smooth Maps
- 3 Lifting Geometry to Mapping Spaces I: Lie Groups
- 4 Lifting Geometry to Mapping Spaces II: (Weak) Riemannian Metrics
- 5 Weak Riemannian Metrics with Applications in Shape Analysis
- 6 Connecting Finite-Dimensional, Infinite-Dimensional and Higher Geometry
- 7 Euler–Arnold Theory: PDEs via Geometry
- 8 The Geometry of Rough Paths
- Appendix A A Primer on Topological Vector Spaces and Locally Convex Spaces
- Appendix B Basic Ideas from Topology
- Appendix C Canonical Manifold of Mappings
- Appendix D Vector Fields and Their Lie Bracket
- Appendix E Differential Forms on Infinite-Dimensional Manifolds
- Appendix F Solutions to Selected Exercises
- References
- Index
2 - Spaces and Manifolds of Smooth Maps
Published online by Cambridge University Press: 08 December 2022
- Frontmatter
- Contents
- Preface
- 1 Calculus in Locally Convex Spaces
- 2 Spaces and Manifolds of Smooth Maps
- 3 Lifting Geometry to Mapping Spaces I: Lie Groups
- 4 Lifting Geometry to Mapping Spaces II: (Weak) Riemannian Metrics
- 5 Weak Riemannian Metrics with Applications in Shape Analysis
- 6 Connecting Finite-Dimensional, Infinite-Dimensional and Higher Geometry
- 7 Euler–Arnold Theory: PDEs via Geometry
- 8 The Geometry of Rough Paths
- Appendix A A Primer on Topological Vector Spaces and Locally Convex Spaces
- Appendix B Basic Ideas from Topology
- Appendix C Canonical Manifold of Mappings
- Appendix D Vector Fields and Their Lie Bracket
- Appendix E Differential Forms on Infinite-Dimensional Manifolds
- Appendix F Solutions to Selected Exercises
- References
- Index
Summary
In this chapter, we consider spaces of differentiable mappings as infinite-dimensional spaces. These spaces will then serve as the model spaces for manifolds of mappings, i.e. manifolds of differentiable mappings between manifolds. The resulting manifolds will allow us to construct essential examples in later chapters, such as the diffeomorphism groups. Moreover, they arise naturally in the context of many applications such as shape analysis and the geometric treatment of partial differential equations which will be discussed later. Finally, we introduce an indispensable tool for the treatment of differentiable mappings on manifolds of mappings in this chapter: the exponential law. roughly speaking, the exponential law allows to interpret a smooth map taking values in a manifold of mappings as a smooth map of two arguments. Thus differentiability questions can often be relegated to a finite dimenisonal setting.
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- Publisher: Cambridge University PressPrint publication year: 2022
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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/cclicenses/
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