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Riemann would have been my guess, too. And I don't see it different than you with the topology and the charts. Nevertheless I can't imagine that the absence of addition or stretching was intended since vector spaces are - although trivial - manifolds, too. But you made me rather curious. Compact, zero curvature, not homeomorphic to a Euclidean space - do you have an example at hand? I'm sure it can be constructed, however, I'm curious whether there is a somehow "common" example.I am not sure of the origin. My point was that this is a question of topology not geometry. But originally curved surfaces may have been the first manifolds considered - not sure. It seems that Riemann first defined manifold in his Habilitation Thesis. The definition is not geometric per se but is topological. It defines the idea of a "multiply extended quantity" - i.e. a space which can be locally coordinatized.