Surfaces | Riemann

: At any point on a Riemann surface, there is a neighborhood homeomorphic to the open unit disk in the complex plane.

: When two such neighborhoods overlap, the "transition maps" between them must be bi-holomorphic (analytic with an analytic inverse), ensuring the complex structure is consistent. Riemann Surfaces

: One of the primary historical motivations for these surfaces was to turn multivalued functions —like the square root or logarithm—into single-valued ones by "lifting" their domain onto multiple connected "sheets". Common Examples Riemann Surfaces : At any point on a Riemann surface,