Here is a set of excerpts from Doolin and Martin (1990), Introduction to Differential Geometry for Engineers, that I used in discussing differentiable manifolds with other participants in the IEEE Standard Upper Ontology Working Group.
What brought the concept of a manifold to mind for me, at the time, was the problem of “interoperability” among “multiple ontologies”. To my way of thinking, that is the very sort of problem that manifolds were invented to handle.
Brian F. Doolin and Clyde F. Martin (1990), Introduction to Differential Geometry for Engineers, Marcel Dekker, New York, NY.
Inquiry Into Inquiry 