Definition:Riemann Surface

A Riemann surface is a $$2 \ $$-dimensional manifold such that every open set $$U \ $$ admits an isomorphism $$\phi:U \to \C \ $$.

A complex manifold is a $$2n \ $$-dimensional manifold such that every open set $$U \ $$ admits an isomorphism $$\phi:U \to \C^n \ $$.