Who discovered Riemann surfaces?
Bernhard Riemann
| Bernhard Riemann | |
|---|---|
| Nationality | German |
| Citizenship | Germany |
| Alma mater | University of Göttingen University of Berlin |
| Known for | See list |
How do you read Riemann surfaces?
A Riemann surface is an oriented manifold of (real) dimension two – a two-sided surface – together with a conformal structure. Again, manifold means that locally at any point x of X, the space is homeomorphic to a subset of the real plane.
Are Riemann surfaces orientable?
Riemann surfaces are always orientable, so in the following review we only consider orientable, triangulable compact surfaces M. We assume that the reader has seen the theory of integration on differentiable manifolds. A Riemann surface is a two dimensional real manifold.
How do you visualize the Riemann sphere?
The Riemann sphere can be visualized as the complex number plane wrapped around a sphere (by some form of stereographic projection – details are given below). In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane, the complex plane plus only one point at infinity.
What is a Riemann surface?
More generally, any meromorphic function can be thought of as a holomorphic function whose codomain is the Riemann sphere. In geometry, the Riemann sphere is the prototypical example of a Riemann surface, and is one of the simplest complex manifolds.
Is the Riemann sphere biholomorphic?
This treatment of the Riemann sphere connects most readily to projective geometry. For example, any line (or smooth conic) in the complex projective plane is biholomorphic to the complex projective line. It is also convenient for studying the sphere’s automorphisms, later in this article.
What is the transition map of a Riemann sphere?
is called the transition map between the two copies of C — the so-called charts — glueing them together. Since the transition maps are holomorphic, they define a complex manifold, called the Riemann sphere. As a complex manifold of 1 complex dimension (i.e., 2 real dimensions), this is also called a Riemann surface .