My mathematics are still quite rudimentary, but am I correct in assuming this is a reference to the "finite" state of closed manifolds as opposed to a potential, "infinite" state of the non-compacted case?
In the soul theorem, published in 1972, Dr. Gromoll and Dr. Cheeger were studying the properties of certain surfaces that could have flat regions or curves like the outside of a sphere but not regions shaped liked saddles. They found that the properties of such surfaces, infinite in extent and existing in any number of dimensions, could be deduced from a finite central core region.
Dr. Cheeger said it was Dr. Gromoll who suggested calling this finite region the “soul” of the object, because it captured the essence of the infinite expanse around it. “Just like inside a person,” Dr. Cheeger said.