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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 ...
This construction is helpful in the study of holomorphic and meromorphic functions. For example, on a compact Riemann surface there are no non-constant holomorphic maps to the complex numbers, but holomorphic maps to the complex projective line are abundant. The Riemann sphere has many uses in physics.
For > the Riemann surface has () parameters (the "moduli"). His contributions to this area are numerous. The famous Riemann mapping theorem says that a simply connected domain in the complex plane is "biholomorphically equivalent" (i.e. there is a bijection between them that is holomorphic with a holomorphic inverse) to either C {\displaystyle ...
This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions. Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture " Ueber die Hypothesen, welche der Geometrie zu Grunde ...
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]
Since every Riemann surface has a universal cover which is a simply connected Riemann surface, the uniformization theorem leads to a classification of Riemann surfaces into three types: those that have the Riemann sphere as universal cover ("elliptic"), those with the plane as universal cover ("parabolic") and those with the unit disk as ...
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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. (Chow's Theorem says that any closed analytic subvariety of projective space is defined by algebraic equations ...