Towards a local definition of body in Continuum Mechanics

Abstract

The purpose of this paper is to introduce the concept of pyramidal manifold, and to demonstrate that it is useful as a model definition of three-dimensional body. Pyramidal manifolds generalize three-dimensional manifolds with corners and represent an approach to the definition of body from the point of view of Differential Geometry, which facilitates the development of the mathematical theory of Continuum Mechanics. Two maps defined on a pyramidal manifold, the degree and the index, are introduced. Both of them are invariant under deformations and allow taking a first step towards a classification of bodies. The Stokes theorem for bodies is also discussed, and a proof there of is provided by using differential forms on pyramidal manifolds.