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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 ...
Any regular surface is an example both of a Riemannian manifold and Riemann surface. Essentially all of the theory of regular surfaces as discussed here has a generalization in the theory of Riemannian manifolds and their submanifolds.
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as smooth manifolds with a Riemannian metric (an inner product on the tangent space at each point that varies smoothly from point to point).
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]
where f satisfies the Cauchy–Riemann equation, and so is holomorphic over its domain. (See Witt algebra.) The conformal isometries of a domain therefore consist of holomorphic self-maps. In particular, on the conformal compactification – the Riemann sphere – the conformal transformations are given by the Möbius transformations
In a simply connected Riemann surface, every closed curve is homotopic to a constant curve for which the integral is zero. Hence a simply connected Riemann surface is planar. If ω is a closed 1-form on a simply connected Riemann surface, ∫ γ ω = 0 for every closed Jordan curve γ. [5] This is the so-called "monodromy property."
Another equivalent definition is as follows: () is the space of pairs (,) where is a Riemann surface and : a diffeomorphism, and two pairs (,), (,) are regarded as equivalent if : is isotopic to a holomorphic diffeomorphism.
In number theory and algebraic geometry, a modular curve Y(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular group of integral 2×2 matrices SL(2, Z).