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:
), 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.