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Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces...

oriented surface are therefore in one-to-one correspondence with conformal classes of metrics on that surface. Within a given conformal class, one can...

geometry of Riemann surfaces. In space higher than two dimensions, conformal geometry may refer either to the study of conformal transformations of what are...

smooth manifolds, distinguished by the fact that the conformal structure on the Riemann surface intrinsically defines a Hodge star operator on 1-forms...

but also determines the Riemann surface up to conformal equivalence. By the uniformization theorem, every compact Riemann surface has simply connected universal...

"Developments of the Theory of Conformal Mapping and Riemann Surfaces Through a Century", Contributions to the Theory of Riemann Surfaces: 3–4 For the...

differential 1-form of compact support is exact. Every simply connected Riemann surface is planar. The class of planar Riemann surfaces was studied by Koebe...

compact Riemann surfaces of genus 1 (the elliptic curves). Complex structures on a closed oriented surface correspond to conformal equivalence classes of Riemannian...

transformations are exactly the bijective conformal maps from the Riemann sphere to itself, i.e., the automorphisms of the Riemann sphere as a complex manifold; alternatively...

makes heavy use of complex analysis. Ahlfors wrote several other significant books, including Riemann surfaces (1960) and Conformal invariants (1973)...

Lorentz surface is a two-dimensional oriented smooth manifold with a conformal equivalence class of Lorentzian metrics. It is the analogue of a Riemann surface...

orientation. Conformal maps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size or curvature. The conformal property...

it is not compact. This can be proved using a version of the Runge theorem for Riemann surfaces, due to Behnke and Stein. Every Stein manifold X {\displaystyle...

T(S)} may be regarded as an isomorphism class of "marked" Riemann surfaces, where a "marking" is an isotopy class of homeomorphisms from S {\displaystyle...

differential geometry of surfaces. Starting with the work of Riemann, the intrinsic point of view was developed, in which one cannot speak of moving "outside"...

system Sasakian manifold Poisson manifold Möbius transformation Conformal map conformal connection tractor bundle Weyl curvature Weyl–Schouten theorem...

geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have...

computer graphics. In the study of Riemann surfaces, one says that any two smooth compact geometric surfaces are "conformally equivalent" when there exists...

Bulletin of the American Mathematical Society. New Series. 23 (2): 566–575. doi:10.1090/S0273-0979-1990-15984-1. Conformal Geometry and Riemann Surfaces: A...

only if they are conformal to one another. For this reason, the study of Riemann surfaces is identical to the study of conformal classes of Riemannian metrics...