An introduction to infinite-dimensional differential geometry / Alexander Schmeding.

Format:

Book

Author/Creator:

Schmeding, A., author.

Series:

Cambridge studies in advanced mathematics ; 202.

Cambridge studies in advanced mathematics ; 202

Language:

English

Subjects (All):

Infinite-dimensional manifolds.

Geometry, Differential.

Physical Description:

1 online resource (xiv, 267 pages) : digital, PDF file(s).

Edition:

First edition.

Place of Publication:

Cambridge, United Kingdom ; New York, NY : Cambridge University Press, 2023.

Summary:

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.

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.

Notes:

Title from publisher's bibliographic system (viewed on 12 Dec 2022).

Includes bibliographical references and index.

ISBN:

9781009089302

1009089307

9781009091251

1009091255

OCLC:

1359047828

Access Restriction:

Open Access. Unrestricted online access