An Introduction to Infinite-Dimensional Differential Geometry

✍ Scribed by Alexander Schmeding

• Publisher: Cambridge University Press
• Year: 2022
• Tongue: English
• Leaves: 284
• Series: Cambridge Studies in Advanced Mathematics, 202
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Cover
Half-title page
Series page
Title page
Copyright page
Contents
Preface
Conventions
Recommended Further Reading
Acknowledgements
1 Calculus in Locally Convex Spaces
1.1 Introduction
1.2 Curves in Locally Convex Spaces
1.3 Bastiani Calculus
1.4 Bastiani versus Fréchet Calculus on Banach Spaces
1.5 Infinite-Dimensional Manifolds
1.6 Tangent Spaces and the Tangent Bundle
1.7 Elements of Differential Geometry: Submersions and Immersions
1.7.1 Exercises
2 Spaces and Manifolds of Smooth Maps
2.1 Topological Structure of Spaces of Differentiable Mappings
2.2 The Exponential Law and Its Consequences
2.3 Manifolds of Mappings
3 Lifting Geometry to Mapping Spaces I: LieGroups
3.1 (Infinite-Dimensional) Lie Groups
3.2 The Lie Algebra of a Lie Group
3.3 Regular Lie Groups and the Exponential Map
3.4 The Current Groups
Loop Groups
Groups of Gauge Transformations
4 Lifting Geometry to Mapping Spaces II: (Weak) Riemannian Metrics
4.1 Weak and Strong Riemannian Metrics
4.2 The Geodesic Distance on a Riemannian Manifold
Geodesics on Infinite-dimensional Manifolds (Informal Discussion)
4.3 Geodesics, Sprays and Covariant Derivatives
Covariant Derivatives
Weak Riemannian Metrics with and without Metric Derivative
4.4 Geodesic Completeness and the Hopf–Rinow Theorem
5 Weak Riemannian Metrics with Applications in Shape Analysis
5.1 The L[sup(2)]-metric and Its Cousins
5.2 Shape Analysis via the Square Root Velocity Transform
6 Connecting Finite-Dimensional, Infinite-Dimensional and Higher Geometry
6.1 Diffeomorphism Groups Determine Their Manifolds
6.2 Lie Groupoids and Their Bisections
6.3 (Re-)construction of a Lie Groupoid from Its Bisections
7 Euler–Arnold Theory: PDEs via Geometry
7.1 Introduction
7.2 The Euler Equation for an Ideal Fluid
7.3 Euler–Poincaré Equations on a Lie Group
7.4 An Outlook on Euler–Arnold Theory
8 The Geometry of Rough Paths
8.1 Introduction
8.2 Iterated Integrals and the Tensor Algebra
8.3 A Rough Introduction to Rough Paths
8.4 Rough Paths and the Shuffle Algebra
8.5 The Grand Geometric Picture (Rough Paths and Beyond)
Appendix A A Primer on Topological Vector Spaces and Locally Convex Spaces
A.1 Basic Material on Topological Vector Spaces
A.2 Seminorms and Convex Sets
A.3 Subspaces of Locally Convex Spaces
A.4 On Smooth Bump Functions
A.5 Inverse Function Theorem beyond Banach Spaces
A.6 Differential Equations beyond Banach Spaces
A.7 Another Approach to Calculus: Convenient Calculus
Bastiani versus Convenient Calculus
Appendix B Basic Ideas from Topology
B.1 Initial and Final Topologies
B.2 The Compact Open Topology
Appendix C Canonical Manifold of Mappings
C.1 Local Additions
C.2 Vector Bundles and Their Sections
C.3 Construction of the Manifold Structure
C.4 Manifolds of Curves and the Energy of a Curve
Appendix D Vector Fields and Their Lie Bracket
D.1 Construction
Appendix E Differential Forms on Infinite-Dimensional Manifolds
E.1 Introduction
E.2 The Maurer–Cartan Form on a Lie Group
E.3 Supplement: Volume Form and Classical Differential Operators
Classical Differential Operators on a Riemannian Manifold
Appendix F Solutions to Selected Exercises
References
Index