In mathematics, a compact Riemann surface is a complex manifold of dimension one that is a compact space. Riemann surfaces are generally classified first into the compact (those that are closed manifolds) and the open (the rest, which from the point of view of complex analysis are very different, being for example Stein manifolds).
A compact Riemann surface C that is a connected space is an algebraic curve defined over the complex number field. More precisely, the meromorphic functions on C make up the function field F on the corresponding curve; F is a field extension of the complex numbers of transcendence degree equal to 1. It can in fact be generated by two functions f and g. This is a structural result on the meromorphic functions: there are enough in the sense of separating out the points of C, and any two are algebraically dependent. These facts were known in the nineteenth century (see GAGA for more in this direction).
A general compact Riemann surface is therefore a finite disjoint union of complex (non-singular) algebraic curves.
Famous quotes containing the words compact and/or surface:
“What compact mean you to have with us?
Will you be pricked in number of our friends,
Or shall we on, and not depend on you?”
—William Shakespeare (1564–1616)
“Puritanism, in whatever expression, is a poisonous germ. On the surface everything may look strong and vigorous; yet the poison works its way persistently, until the entire fabric is doomed.”
—Emma Goldman (1869–1940)