One-forms on spaces of embeddings: a frame work for constitutive laws in elasticity

Abstract

The present contribution to this volume is concerned with certain problems in non-linear functional analysis which are motivated by classical physics, specifically by elasticity theory:we are given a «body»,i.e. a compact smooth manifold $M'$ which moves and may be deformed in some $Rⁿ$ (equipped with a fixed inner product); we assume that the motion and deformation are such that the diffeomorphism type of $M'$ does not change. Hence, $M'$ is the image under a smooth embedding of some compact smooth manifold M (possibly with boundary $\partial M$) and the appropriate configuration space for the problem is the set $E( M, Rⁿ)$ of smooth embeddings $M→ Rⁿ$; this set is a smooth Fréchet manifold when endowed with its natural $C^∈fty$-topology.