A sense of how manifolds are applied in practice may be gleaned from the set of excerpts linked below, from Doolin and Martin (1990), Introduction to Differential Geometry for Engineers, which I used in discussing differentiable manifolds with other participants in the IEEE Standard Upper Ontology Working Group.

• Differential Geometry for Engineers • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9)

What brought the concept of a manifold to mind in that context was a set of problems associated with perspectivity, relativity, and interoperability among multiple ontologies.  To my way of thinking, those are the very sorts of problems manifolds were invented to handle.

Reference

• Doolin, Brian F., and Martin, Clyde F. (1990), Introduction to Differential Geometry for Engineers, Marcel Dekker, New York, NY.