Riemann surfaces, two dimensional manifolds equipped with a complex structure, which are topologically classified by the genus, are an important example of this phenomenon.
In 1857, Riemann introduced the concept of Riemann surfaces as part of a study of the process of analytic continuation; Riemann surfaces are now recognized as one-dimensional complex manifolds.
The idea of a Grothendieck topology (also known as a site) has been characterised by John Tate as a bold pun on the two senses of Riemann surface.
Note that multi-valued functions such as the natural logarithm may be used, but attention must be confined to a single Riemann surface.
In mathematics, a Veech surface is a translation surface (X,ω) (a Riemann surface X with a holomorphic 1-form ω) whose group SL(X,ω) of affine diffeomorphisms is a lattice in SL2(R) (a discrete subgroup of cofinite volume).
surface-to-air missile | Bernhard Riemann | Surface-to-air missile | Surface | Riemann zeta function | Riemann surface | Riemann hypothesis | Surface (TV series) | surface | Fermi surface | Surface-mount technology | Riemann sphere | Naval Surface Warfare Center Crane Division | Naval Surface Warfare Center | Microsoft Surface | Type 91 Surface-to-air missile | Surface weather observation | Surface wave magnitude | Surface Water Ocean Topography Mission | Surface Warfare insignia | Surface texture | surface area | Shioda modular surface | Road surface | Riemann solver | Riemann-Roch theorem | Mars surface color | Kummer surface | Intermodal Surface Transportation Efficiency Act | Hugo Riemann |
The system considers the motion of a free (frictionless) particle on a surface of constant negative curvature, the simplest compact Riemann surface, which is the surface of genus two: a donut with two holes.
In its original form, it was applied to an unramified double covering of a Riemann surface, and was used by F. Schottky and H. W. E. Jung in relation with the Schottky problem, as it now called, of characterising Jacobian varieties among abelian varieties.
The theorem for compact Riemann surfaces can be deduced from the algebraic version using Chow's theorem and the GAGA principle: in fact, every compact Riemann surface is defined by algebraic equations in some complex projective space.