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The motivating force of Topology, consisting of the study of smooth (differentiable) Manifolds. Differential topology deals with nonmetrical notions of Manifolds, while Differential Geometry deals with metrical notions of Manifolds.
See also Differential Geometry
References
Dieudonné, J. A History of Algebraic and Differential Topology: 1900-1960. Boston, MA:
Birkhäuser, 1989.
Munkres, J. R. Elementary Differential Topology. Princeton, NJ: Princeton University Press, 1963.