Is Riemann sphere a Riemann surface?

In geometry, the Riemann sphere is the prototypical example of a Riemann surface, and is one of the simplest complex manifolds.

How many Riemann sheets are there?

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. On the sheet R0, let the angles θ1, θ2 range from 0 to 2π.

What is the theory of moduli spaces of Riemann surfaces?

These moduli spaces are the geometric solution to the problem of classification of compact Riemann surfaces, and can be thought of as the “higher theory” of Riemann surfaces. The moduli spaces are “mean- ingful spaces,” in that each of their points stands for a Riemann surface.

Who discovered Riemann surfaces?

A brief history of the subject: Riemann surfaces were first defined in Riemann’s 1851 PhD dissertation [7] where he laid down the geometric foundations of the theory of functions of one complex variable.

Is a Riemann surface connected?

There are several equivalent definitions of a Riemann surface. A Riemann surface X is a connected complex manifold of complex dimension one. Again, manifold means that locally at any point x of X, the space is homeomorphic to a subset of the real plane.

What is complex infinity?

Complex infinity is an infinite number in the complex plane whose complex argument is unknown or undefined. Complex infinity may be returned by the Wolfram Language, where it is represented symbolically by ComplexInfinity. The Wolfram Functions Site uses the notation. to represent complex infinity.

What is a branch cut?

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. Instead, lines of discontinuity must occur. The most common approach for dealing with these discontinuities is the adoption of so-called branch cuts.

Is a torus a Riemann surface?

Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together.

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.

What is infinity in complex plane?

By definition the extended complex plane =C∪{∞}. That is, we have one point at infinity to be thought of in a limiting sense described as follows. A sequence of points {zn} goes to infinity if |zn| goes to infinity. This “point at infinity” is approached in any direction we go.

Is infinity a number?

Infinity is not a number. Instead, it’s a kind of number. You need infinite numbers to talk about and compare amounts that are unending, but some unending amounts—some infinities—are literally bigger than others. When a number refers to how many things there are, it is called a ‘cardinal number’.

Is the holomorphic function on a Riemann surface constant?

Every non-compact Riemann surface admits non-constant holomorphic functions (with values in C ). In fact, every non-compact Riemann surface is a Stein manifold . In contrast, on a compact Riemann surface X every holomorphic function with values in C is constant due to the maximum principle.

What kind of functions can be found on a Riemann surface?

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 .

Can a compact Riemann surface be embedded in a projective space?

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.

Can a hyperbolic Riemann surface be an orientable surface?

Hyperbolic Riemann surfaces. In the remaining cases is a hyperbolic Riemann surface, that is isomorphic to a quotient of the upper half-plane by a Fuchsian group (this is sometimes called a Fuchsian model for the surface). The topological type of can be any orientable surface save the torus and sphere.