On Families in Differential Geometry
arXiv:1302.2061 · doi:10.1142/S0219887813500424
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.
19 pages. Accepted for publication on International Journal of Geometric Methods in Modern Physics (8-2-2013)