Each simply connected, open subset of one of these is holomorphic to either the complex plane, projective complex plane, or unit open disc, and conversely, according to the uniformization theorem.
Each simply connected, open subset of one of these is holomorphic to either the complex plane, projective complex plane, or unit open disc, and conversely, according to the uniformization theorem. The number of automorphisms on one of these is bounded by the Hurwitz theorem if its genus is at least one. Holomorphic functions between two of these spaces that are equal on a subset of the domain are generally equal everywhere according to these spaces' namesake identity theorem. An alternative way to understand these is to introduce a branch cut into the complex plain, which can then be viewed as the joining of the sheets of one of these; moving between sheets of one of these is then equivalent to switching analytic continuations of a function. For 10 points, name these real, 2-dimensional manifolds that arise naturally when considering how to unambiguously define multi-valued functions.
Each simply connected, open subset of one of these is holomorphic to either the complex plane, projective complex plane, or unit open disc, and conversely, according to the uniformization theorem.
Riemann surfaces