# Riemann surface A Riemann surface is a 2D [[Topological manifold (manifold)|manifold]] with complex structure. A smooth manifold is a space that can be covered by flat coordinate maps, and whenever two maps overlap, the change of coordinates is smooth. This enable calculus to be consistent over the space. It also mean that the calculus will be independent of the coordination chosen in each map. $\mathbb{C}$-structure (complex structure) means that the manifold maps are subspace of the complex plane $\mathbb{C}$ and the transition maps are holomorphic (complex smoothness). Example: 2D-sphere $S^2$ can be made a Riemann surface by identifying it with the Riemann sphere $\widehat{\mathbb{C}}:=\mathbb{C}\cup\{\infty\}$ . The 1st map sends points from the sphere $S^2$ (except the north pole) to the plain $\mathbb{C}$ via line-projection from the north pole. The 2nd map do the same from the south pole.The transition map is $f(z)=\frac{1}{z}$ which is holomorphic. Counter example Mobius strip: a 2D manifold that support complex structure is orientable. As Mobius strip is not orientable it does not allow complex structure and so it is not a Riemann surface. ## Created 2026-05-29 15:10