An Introduction to Infinite-Dimensional Differential Geometry
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
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

An Introduction to Infinite-Dimensional Differential Geometry

by Alexander Schmeding
An Introduction to Infinite-Dimensional Differential Geometry

An Introduction to Infinite-Dimensional Differential Geometry

by Alexander Schmeding

Overview

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.

Product Details

ISBN-13: 9781316514887
Publisher: Cambridge University Press
Publication date: 12/22/2022
Series: Cambridge Studies in Advanced Mathematics , #202
Pages: 280
Product dimensions: 6.22(w) x 9.25(h) x 0.91(d)

About the Author

Alexander Schmeding is Associate Professor in Mathematics at Nord University at Levanger.

Table of Contents

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: PDE via geometry; 8. The geometry of rough paths; A. A primer on topological vector spaces and locally convex spaces; B. Basic ideas from topology; C. Canonical manifold of mappings; D. Vector fields and their Lie bracket; E. Differential forms on infinite-dimensional manifolds; F. Solutions to selected exercises; References; Index.
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