An Introduction to Infinite-Dimensional Differential Geometry

Alexander Schmeding (Nord Universitet, Norway)

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English

Cambridge University Press
22 December 2022

Geometry; Mathematical physics

Series: Cambridge Studies in Advanced Mathematics

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.

By: Alexander Schmeding (Nord Universitet Norway)
Imprint: Cambridge University Press
Country of Publication: United Kingdom
Dimensions: Height: 235mm, Width: 158mm, Spine: 23mm
Weight: 570g
ISBN: 9781316514887
ISBN 10: 1316514889
Series: Cambridge Studies in Advanced Mathematics
Pages: 280
Publication Date: 22 December 2022
Audience: General/trade , ELT Advanced
Format: Hardback
Publisher's Status: Active

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.

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