An Introduction to Infinite-Dimensional Differential Geometry

Abstract

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.

Manifolds of mappings on Cartesian products

“…We use C m -maps between open subsets of locally convex spaces in the sense of Bastiani [Bas64] and the corresponding C ∞ -manifolds and Lie groups (see [Glö02a; GN24; Nee06; Sch23] for further information; cf. also [Ham82;Mic80;Mil84;Nee06;Sch23]). Thus, manifolds and Lie groups are modeled on locally convex spaces which can be infinite dimensional, unless the contrary is stated.…”

“…The map ρ : Diff fr (M ) → im(ρ) discussed in Theorem 1.1 is a smooth group homomorphism between Lie groups and a submersion. Hence L(ρ) = T e (ρ) has a continuous linear right inverse (see[Sch23, p. 1.56]). Since Diff fr (M ) is L 1 -regular (cf.…”

Covariant canonical formulations of classical field theories

“…→ E is smooth if and only if λ • γ is smooth for all continuous linear functionals λ on E. For the particular case of Fréchet spaces like C ∞ ( ), the convenient calculus coincides with the Gateaux approach to differentiation, but for more general model vector spaces it yields a notion of smooth map which does not necessarily imply continuity. For the relationship between different definitions of infinite-dimensional manifolds and notions of smoothness (in particular the Bastiani calculus which was introduced by Andrée Bastiani in her PhD thesis [364] and which is considered in works of field theory [11,145]), we refer to [343,356,365,366] and the appendix of [367].…”

A multiplicity theorem for Fréchet spaces

“…However, expanding beyond the Banach setting necessitates a robust concept of submersion to extend the typical assertions regarding submersion to manifolds modeled over locally convex spaces. For such manifolds, various non-equivalent mappings, including infinitesimally surjective, naïve submersion, and submersion, are available for constructing submanifolds, see [20]. In [9], submersions have been utilized to construct submanifolds in the case of manifolds modeled on locally convex spaces.…”