An introduction to infinite-dimensional differential geometry

Abstract

The present document is the draft of a book which presents an introduction to infinite-dimensional differential geometry beyond Banach manifolds. As is well known the usual calculus breaks down in this setting. Hence, we replace it by the more general Bastiani calculus which is built using directional derivatives. We then focus on two main areas of infinite-dimensional geometry: 1. infinite-dimensional Lie groups, and 2. weak Riemannian geometry. Both topics are developed and connected to manifolds of (smooth) mappings. These manifolds are studied in detail to construct important examples such as diffeomorphism groups, loop groups and Riemannian metrics for shape analysis. Manifolds of mappings are prime examples for surprising connections between finite and infinite-dimensional geometry. However, also pathologies occurring in infinite-dimensions will be highlighted in many examples. The geometric techniques developed will then be showcased in modern applications of geometry such as geometric hydrodynamics, higher geometry in the guise of Lie groupoids and rough path theory.

260 pages, 2 figures, uses TikZ, comments welcome, draft for a book to be published by Cambridge University Press, v3: Improved exposition in chapter 8, corrected many typos and smaller misprints, rest unchanged

References in corpus (6)

• Geodesic Equations on Diffeomorphism Groups
• Lie Group Structures on Symmetry Groups of Principal Bundles
• How to Construct a Dirac Operator in Infinite Dimensions
• The Geometry of Loop Spaces I: $H^s$-Riemannian Metrics
• Finslerian geodesics on Fréchet manifolds
• Manifolds of mappings on cartesian products

Cited by in corpus (7)

• Covariant canonical formulations of classical field theories
• Lie Theory for Asymptotic Symmetries in General Relativity: The NU Group
• A symplectic approach to Schrödinger equations in the infinite-dimensional unbounded setting
• Group Contractions via Infinite-Dimensional Lie Theory
• Controllability and diffeomorphism groups on manifolds with boundary
• Manifolds of continuous BV-functions and vector measure regularity of Banach-Lie groups
• The Michor-Mumford conjecture in Hilbertian H-type groups