Manifolds of mappings for continuum mechanics

TL;DR

This paper provides an overview of the mathematical structure of manifolds of smooth mappings between manifolds, emphasizing their foundational role in geometric continuum mechanics.

Contribution

It introduces the concept of Whitney manifold germs and develops the manifold structure of smooth mappings for applications in continuum mechanics.

Findings

Defines Whitney manifold germs with boundary conditions

Establishes the manifold structure of $C^{ abla}(M,N)$

Provides foundational tools for geometric continuum mechanics

Abstract

This is an overview article. After an introduction to convenient calculus in infinite dimensions, the foundational material for manifolds of mappings is presented. The central character is the smooth convenient manifold $C^{\infty}(M,N)$ of all smooth mappings from a finite dimensional Whitney manifold germ $M$ into a smooth manifold $N$. A Whitney manifold germ is a smooth (in the interior) manifold with a very general boundary, but still admitting a continuous Whitney extension operator. This notion is developed here for the needs of geometric continuum mechanics.