Riemann Surface - Examples

Examples

• The complex plane C is the most basic Riemann surface. The map f(z) = z (the identity map) defines a chart for C, and {f} is an atlas for C. The map g(z) = z* (the conjugate map) also defines a chart on C and {g} is an atlas for C. The charts f and g are not compatible, so this endows C with two distinct Riemann surface structures. In fact, given a Riemann surface X and its atlas A, the conjugate atlas B = {f* : f ∈ A} is never compatible with A, and endows X with a distinct, incompatible Riemann structure.

• In an analogous fashion, every open subset of the complex plane can be viewed as a Riemann surface in a natural way. More generally, every open subset of a Riemann surface is a Riemann surface.

• Let S = C ∪ {∞} and let f(z) = z where z is in S \ {∞} and g(z) = 1 / z where z is in S \ {0} and 1/∞ is defined to be 0. Then f and g are charts, they are compatible, and { f, g } is an atlas for S, making S into a Riemann surface. This particular surface is called the Riemann sphere because it can be interpreted as wrapping the complex plane around the sphere. Unlike the complex plane, it is compact.

• The theory of compact Riemann surfaces can be shown to be equivalent to that of projective algebraic curves that are defined over the complex numbers and non-singular. For example, the torus C/(Z + τ Z), where τ is a complex non-real number, corresponds, via the Weierstrass elliptic function associated to the lattice Z + τ Z, to an elliptic curve given by an equation

y2 = x3 + a x + b.

Tori are the only Riemann surfaces of genus one, surfaces of higher genera g are provided by the hyperelliptic surfaces

y2 = P(x),

where P is a complex polynomial of degree 2g + 1.

• Important examples of non-compact Riemann surfaces are provided by analytic continuation.

• f(z) = arcsin z

• f(z) = log z

• f(z) = z1/2

• f(z) = z1/3

• f(z) = z1/4