# Are Riemann surfaces orientable?

## Are Riemann surfaces orientable?

Each Riemann surface, being a complex manifold, is orientable as a real manifold. For complex charts f and g with transition function h = f(g−1(z)), h can be considered as a map from an open set of R2 to R2 whose Jacobian in a point z is just the real linear map given by multiplication by the complex number h'(z).

## Is E z multivalued function?

Moreover, eq. (2) implies that arg ez = Arg ez + 2πn, where n = 0, ±1, ±2,…. Hence, arg(ez) = y + 2πk, where k = n + N is still some integer. This is not surprising, since ln(ez) is a multi-valued function, which cannot be equal to the single-valued function z.

How do you know if a function is multivalued?

Examples

1. Every real number greater than zero has two real square roots, so that square root may be considered a multivalued function.
2. Each nonzero complex number has two square roots, three cube roots, and in general n nth roots.
3. The complex logarithm function is multiple-valued.

### Is the Riemann sphere a Riemann surface?

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.

### Is exp z multivalued?

Here lnz is defined by exp(lnz)=z and is multi-valued. ln is multi valued, exponential is not.

Why is a multivalued function a function?

A multivalued function, also known as a multiple-valued function (Knopp 1996, part 1 p. 103), is a “function” that assumes two or more distinct values in its range for at least one point in its domain.

## Why is the Riemann sphere important?

The Riemann Sphere is a fantastic glimpse of where geometry can take you when you escape from the constraints of Euclidean Geometry – the geometry of circles and lines taught at school. Riemann, the German 19th Century mathematician, devised a way of representing every point on a plane as a point on a sphere.

## Can a function be multivalued?

Is a multivalued function analytic?

Analytic Functions Multivalued functions can also be analytic under certain restrictions that make them single-valued in specific regions; this case, which is of great importance, is taken up in detail in Section 11.6. If f (z) is analytic everywhere in the (finite) complex plane, we call it an entire function.

### Is infinity a point?

Though a point at infinity is considered on a par with any other point of a projective range, in the representation of points with projective coordinates, distinction is noted: finite points are represented with a 1 in the final coordinate while a point at infinity has a 0 there.

### What is the most advanced geometry?

The most advanced part of plane Euclidean geometry is the theory of the conic sections (the ellipse, the parabola, and the hyperbola).

What is the basic idea of Riemann surface theory?

Thus, the basic idea of Riemann surface theory is to replace the domain of a multi-valued function, e.g. a function de\fned by a polynomial equation P(z;w) = wn+ p n 1(z)wn 1+ + p 1(z)w+ p 0(z) by its graph S= f(z;w) 2C2jP(z;w) = 0g; and to study the function was a function on the ‘Riemann surface’ S, rather than as a multi- valued function of z.

## Is the quotient h=Sl (2;Z) a Riemann surface?

In other words, the quotient H=SL(2;Z) inherits the structure of an abstract Riemann surface; and Jestablishes an analytic isomorphism between this surface and C. Lecture 12 Two new Weierstraˇ functions

## How to use the Riemann theorem on the local form?

To use the theorem on the local form with more ease, observe the following. 3.12 Lemma: If a holomorphic map between Riemann surfaces is constant in a neighbourhood of a point, then it is constant on a connected component of that surface which contains the point.

How do you find the Riemann surface of a polynomial?

Consider the Riemann surface of the equation P(z;w) = 0 for a polynomial P(z;w) = wn+ p n 1(z)wn 1+ + p 1(z)w+ p