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There are several equivalent definitions of a Riemann surface. A Riemann surface X is a connected complex manifold of complex dimension one. This means that X is a connected Hausdorff space that is endowed with an atlas of charts to the open unit disk of the complex plane: for every point x ∈ X there is a neighbourhood of x that is homeomorphic to the open unit disk of the complex plane, and ...
Colloquially speaking, the genus of a Riemann surface is its number of handles; for example the genus of the Riemann surface shown at the right is three. More precisely, the genus is defined as half of the first Betti number , i.e., half of the C {\displaystyle \mathbb {C} } -dimension of the first singular homology group H 1 ( X , C ...
If a complete n-dimensional Riemannian manifold has nonnegative Ricci curvature and a straight line (i.e. a geodesic that minimizes distance on each interval) then it is isometric to a direct product of the real line and a complete (n-1)-dimensional Riemannian manifold that has nonnegative Ricci curvature. Bishop–Gromov inequality.
In mathematics, the Riemann–Roch theorem for surfaces describes the dimension of linear systems on an algebraic surface. The classical form of it was first given by Castelnuovo (1896, 1897), after preliminary versions of it were found by Max Noether and Enriques . The sheaf-theoretic version is due to Hirzebruch.
The fundamental objects are called the Riemannian metric and the Riemann curvature tensor. For the surface (two-dimensional) case, the curvature at each point can be reduced to a number (scalar), with the surfaces of constant positive or negative curvature being models of the non-Euclidean geometries.
Riemannian manifolds were first conceptualized by their namesake, German mathematician Bernhard Riemann.. In 1827, Carl Friedrich Gauss discovered that the Gaussian curvature of a surface embedded in 3-dimensional space only depends on local measurements made within the surface (the first fundamental form). [1]
On a Riemann surface the Poincaré lemma states that every closed 1-form or 2-form is locally exact. [2] Thus if ω is a smooth 1-form with dω = 0 then in some open neighbourhood of a given point there is a smooth function f such that ω = df in that neighbourhood; and for any smooth 2-form Ω there is a smooth 1-form ω defined in some open neighbourhood of a given point such that Ω = dω ...
Riemann was the first one to do extensive work generalizing the idea of a surface to higher dimensions. The name manifold comes from Riemann's original German term, Mannigfaltigkeit, which William Kingdon Clifford translated as "manifoldness".