The quotient space T \ H can be identified with a compact

Riemann surface F of a genus g [greater than or equal to] 2.

In order to acquire the algebro-geometric solutions of systems (16), we first introduce the

Riemann surface r of the hyperelliptic curve with genus N:

[4]), and they found that the number of minimal reducing subspaces of [T.sub.B] equals the number of connected components of the

Riemann surface of B(z) = B(w) when the order of B is 3,4,6.

Consider a compact connected

Riemann surface [summation] of genus g with canonical homology cycle bases [a.sub.i], [b.sub.i] for i = 1, ..., g.

Along the lines of other important contributions to the development of topology--and in particular that of Felix Klein--Hermann Weyl has pointed out this feature of Riemann's work, by arguing "that it is always the

Riemann surface, not the analytic form, which is regarded as the given object" (2010: 157).

Let [GAMMA]\H be a hyperbolic

Riemann surface of finite area.

In the case of multi-valued analytic functions of a single complex variable, the proposed solution is called "The

Riemann Surface'.

Hence if, conversely, a map A from a simply connected

Riemann surface [summation] into ([h.sup.2n+1]) satisfying (3.5) is given, then a solution to [[phi].sup.-1]ld[phi] = Bdz + [bar.B]d[bar.z] exists and defines a harmonic map from [summation] into [H.sup.2n+1].

Another interesting instance is that of a

Riemann surface endowed with the Poincare metric.

The function g[lambda] has branch points at [z.sup.+.sub.[lambda]] and [z.sup.-.sub.[lambda]] and its natural domain of definition is the two-sheeted

Riemann surface [R.sub.[lambda]] defined by the relation

For any

Riemann surface S of genus g [less than or equal to] 2,

This distinction is necessary because [u.sup.(p).sub.s] and [[partial]u.sup.(s).sub.s]/[partial]z are discontinuous on crossing the screen, and is taken into account in Sommerfeld's theory by "wrapping" the diffracting half plane in a semi-infinite, two-sided

Riemann surface so that its positive and negative sides are distinguished by the values 2[pi] and 0 of the polar angle [PHI] in Fig.