An Introduction to Infinite-Dimensional Differential Geometry

This text introduces infinite-dimensional differential geometry beyond Banach manifolds, focusing on Lie groups and weak Riemannian geometry. It explores connections to manifolds of (smooth) mappings, with examples in modern applications such as geometric hydrodynamics and higher geometry. It is essential for graduate students and researchers in functional analysis, non-linear differential equations, and differential geometry.

Format: Hardback
Length: 280 pages
Publication date: 22 December 2022
Publisher: Cambridge University Press

Beyond Banach manifolds, this text delves into foundational concepts in infinite-dimensional differential geometry, drawing from Bastiani calculus. It focuses on two primary areas of infinite-dimensional geometry: infinite-dimensional Lie groups and weak Riemannian geometry. It explores the connections between these geometries and manifolds of (smooth) mappings. The book covers topics such as diffeomorphism groups, loop groups, and Riemannian metrics for shape analysis. Numerous examples illustrate both surprising connections between finite- and infinite-dimensional geometry and the unique challenges that arise in infinite dimensions. The geometric techniques developed are then applied to modern applications, including geometric hydrodynamics, higher geometry in the form of Lie groupoids, and rough path theory. With ample exercises, some with solutions, and worked examples, this book is an essential resource for graduate students and researchers working at the intersection of functional analysis, non-linear differential equations, and differential geometry. Additionally, it is available as Open Access on Cambridge Core.

ISBN-13: 9781316514887