Riemann formed, by extension, a multiconnected many-sheeted surface that could be dissected by cross-cuts into a singly connected surface. By means of these surfaces he introduced topological considerations into the theory of functions of a complex variable, and into general analysis.

*Feb 01, 2013 · One of the clearest books I know of to learn the topics in its title. I used this for my last course on Riemann surfaces and algebraic curves in 2010. I learned a lot myself and thoroughly enjoyed the reading. Good exercises too ...*!2 has a preferred branch at the origin, determined by the standard branch of logarithm. In this way, we see that !2 is a canonically de ned, single-valued, holomorphic one-form on the Riemann surface T 2. This Riemann-surface has in nitely many sheets over our original space T. Continuing in this way to compute the next power, we nd!3 = Li 2 1 ... of a periodic minimal surface in R3 can usually be described in terms of the geometry of its quotient surface M in the at three manifold R3=. Thus a triply periodic minimal surface is a minimal surface in a at 3-torus R3=. the canonical sheaf of an algebraic variety. The genus It is well known that for a closed and orientable topological surface S, the genus represents what one intuitively would call \the number of handles". This concept has many interpretations and ways to be de ned. From a topological perspective, we can