# 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

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