Riemann Surface
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A Riemann Surface is a Complex Manifold that provides a way to extend the methods of complex analysis.
- Context:
- It can (often) serve as a Complex Curves Model, simplifying the analysis of holomorphic functions.
- It can range from being a Compact Riemann Surface to being a Non-Compact Riemann Surface.
- It can be classified by its genus, which indicates the surface's number of "holes".
- It can represent the domain where analytic continuation of complex functions is possible.
- It can illustrate solutions to problems in complex dynamics, algebraic geometry, and other fields.
- It can be covered by a collection of coordinate charts where transition maps are holomorphic.
- ...
- Example(s):
- a Complex Plane that showcases the simplest case of a Riemann Surface.
- a Riemann Sphere, which demonstrates a compact Riemann Surface.
- an Elliptic Curve, which is a torus and also a Riemann Surface of genus 1.
- a Hyperbolic Riemann Surface, which represents a surface with constant negative curvature.
- ...
- Counter-Example(s):
- Real Manifolds, which do not necessarily support complex structures.
- Affine Varietys, which are not necessarily equipped with the same complex analytical properties.
- See: Complex Analysis, Holomorphic Function, Genus, Riemann Sphere, Algebraic Geometry.