On Families in Differential Geometry

TL;DR

This paper introduces a unified algebraic framework for families of geometric objects in differential geometry, enabling generalized calculus without ad hoc spaces, exemplified by proving a universal homotopy formula.

Contribution

It develops a coordinate-free algebraic approach to families in differential geometry using differential calculus over commutative algebras, generalizing fundamental calculus theorems.

Findings

Unified framework for families of geometrical quantities

Generalization of fundamental calculus theorems to families

Proof of the universal homotopy formula

Abstract

Families of objects appear in several contexts, like algebraic topology, theory of deformations, theoretical physics, etc. An unified coordinate-free algebraic framework for families of geometrical quantities is presented here, which allows one to work without introducing ad hoc spaces, by using the language of differential calculus over commutative algebras. An advantage of such an approach, based on the notion of sliceable structures on cylinders, is that the fundamental theorems of standard calculus are straightforwardly generalized to the context of families. As an example of that, we prove the universal homotopy formula.