What is the Genus of Riemann Surface?

In summary, a Riemann surface is a mathematical concept that extends the idea of a complex plane to surfaces with multiple connected pieces. The genus of a Riemann surface is determined by the number of handles or "holes" in the surface and is important in complex analysis, algebraic geometry, and physics. Examples of Riemann surfaces include the complex plane, Riemann sphere, and torus, and they are used in various areas of mathematics to solve differential equations, study algebraic curves, and understand complex functions.
  • #1
yicong2011
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I learned something about genus in Topology. The concept Genus in Topology is intuitive and lucid. Now I am confronted with the Genus in Riemann Surface. I do not know what is Genus on Riemann Surface. Is it relevant to "singularity"?

Anyone can help me make it a bit clear?
Thanks.
 
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  • #2
Same thing as genus in topology. A Riemann surface is just a topological surface with extra structure on it. It therefore has a genus.
 
  • #3
here is a set of class notes from my course on RS.
 

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1. What is a Riemann surface?

A Riemann surface is a mathematical concept that extends the idea of a complex plane to surfaces that have multiple connected pieces.

2. How is the genus of a Riemann surface determined?

The genus of a Riemann surface is determined by the number of handles or "holes" in the surface. It can also be calculated using the Euler characteristic, which is 2 - 2g, where g is the genus.

3. What is the significance of the genus of a Riemann surface?

The genus of a Riemann surface is important in complex analysis and algebraic geometry as it provides information about the topological structure of the surface. It also has applications in physics, particularly in string theory.

4. What are some examples of Riemann surfaces?

Some examples of Riemann surfaces include the complex plane, the Riemann sphere, and the torus. Other examples include the hyperbolic plane, elliptic curves, and the modular curve.

5. How are Riemann surfaces used in mathematics?

Riemann surfaces are used in various areas of mathematics, including complex analysis, algebraic geometry, and topology. They are also used in physics, particularly in string theory. Riemann surfaces have applications in solving differential equations, studying algebraic curves, and understanding the behavior of complex functions.

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