An introduction to infinite-dimensional differential geometry

TL;DR

This paper introduces infinite-dimensional differential geometry beyond Banach manifolds, focusing on Bastiani calculus, infinite-dimensional Lie groups, and weak Riemannian geometry, with applications in shape analysis and modern geometric theories.

Contribution

It develops foundational techniques for infinite-dimensional geometry beyond Banach manifolds, connecting geometric structures to applications like shape analysis and hydrodynamics.

Findings

Construction of diffeomorphism and loop groups

Development of weak Riemannian metrics for shape analysis

Identification of pathologies in infinite-dimensional manifolds

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.