When in doubt, differentiate.
Shing-Shen Chern (1979).
The idea of a differentiable manifold had its genesis in the nineteenth century with the work of Carl Friedrich Gauss and of Georg Friedrich Bernhard Riemann. Gauss was interested in surveying and cartography, which led him to develop the tools of calculus on curved surfaces. His famous theorema egregium, or remarkable theorem, revealed that one could consider the intrinsic properties of a surface independently of the way in which it was embedded in three-dimensional space, and this led him, Riemann, and others, to abstract these concepts even further. Their ideas have had far reaching applications in many areas of mathematics and the natural sciences.
Roughly, an n-dimensional manifold (or n-manifold) can be thought of as a kind of patchwork quilt built from pieces of ℝn. Classic examples of 2-manifolds are the 2-sphere S2 and the 2-torus T2 (see Figure 3.1). Usually one pictures these as living in ℝ3, but one can consider them in their own right just as bits of ℝ2 sewn together in certain ways. The technical definition of a manifold requires considerable background, which we will try to keep to a minimum. First, we need the idea of a topology.
Consider a basketball. When it is inflated, its surface is a sphere. But when it is deflated its surface is still a topological sphere.