Table of Contents
What is the use of Riemann surfaces?
The main interest in Riemann surfaces is that holomorphic functions may be defined between them. Riemann surfaces are nowadays considered the natural setting for studying the global behavior of these functions, especially multi-valued functions such as the square root and other algebraic functions, or the logarithm.
Who discovered Riemann surfaces?
Bernhard Riemann
| Bernhard Riemann | |
|---|---|
| Nationality | German |
| Citizenship | Germany |
| Alma mater | University of Göttingen University of Berlin |
| Known for | See list |
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 many sheets are there in the Riemann surface?
A Riemann surface for this function consists of two sheets, R0 and R1. Both sheets are cut along the line segment between ±1. The lower edge of the slit in R0 is joined to the upper edge of the slit in R1, and the lower edge in R1 to the upper edge in R0.
Where did Riemann live?
Germany
Kingdom of Hanover
Bernhard Riemann/Places lived
Bernhard Riemann, in full Georg Friedrich Bernhard Riemann, (born September 17, 1826, Breselenz, Hanover [Germany]—died July 20, 1866, Selasca, Italy), German mathematician whose profound and novel approaches to the study of geometry laid the mathematical foundation for Albert Einstein’s theory of relativity.
What is a branch cut in complex analysis?
A branch cut is a curve (with ends possibly open, closed, or half-open) in the complex plane across which an analytic multivalued function is discontinuous. For convenience, branch cuts are often taken as lines or line segments. Instead, lines of discontinuity must occur.
How do you find the Riemann surface of a function?
Riemann surface for the function f ( z ) = √z. The two horizontal axes represent the real and imaginary parts of z, while the vertical axis represents the real part of √z. The imaginary part of √z is represented by the coloration of the points.
Is every Riemann surface a holomorphic function?
Every Riemann surface is a two-dimensional real analytic manifold (i.e., a surface ), but it contains more structure (specifically a complex structure) which is needed for the unambiguous definition of holomorphic functions.
Can a manifold be turned into a Riemann surface?
A two-dimensional real manifold can be turned into a Riemann surface (usually in several inequivalent ways) if and only if it is orientable and metrizable. So the sphere and torus admit complex structures, but the Möbius strip, Klein bottle and real projective plane do not.
Is every compact Riemann surface a projective variety?
The existence of non-constant meromorphic functions can be used to show that any compact Riemann surface is a projective variety, i.e. can be given by polynomial equations inside a projective space. Actually, it can be shown that every compact Riemann surface can be embedded into complex projective 3-space.