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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 ...
Every Riemann surface is the quotient of the free, proper and holomorphic action of a discrete group on its universal covering and this universal covering, being a simply connected Riemann surface, is holomorphically isomorphic (one also says: "conformally equivalent" or "biholomorphic") to one of the following: the Riemann sphere; the complex ...
A tangent plane of the sphere with two vectors in it. A Riemannian metric allows one to take the inner product of these vectors. Let be a smooth manifold.For each point , there is an associated vector space called the tangent space of at .
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 ...
The concept of a branch point is defined for a holomorphic function ƒ:X → Y from a compact connected Riemann surface X to a compact Riemann surface Y (usually the Riemann sphere). Unless it is constant, the function ƒ will be a covering map onto its image at all but a finite number of points.
A ruled surface is one which can be generated by the motion of a straight line in E 3. [46] Choosing a directrix on the surface, i.e. a smooth unit speed curve c(t) orthogonal to the straight lines, and then choosing u(t) to be unit vectors along the curve in the direction of the lines, the velocity vector v = c t and u satisfy
Broadly, one could analogize the role of the Ricci curvature in Riemannian geometry to that of the Laplacian in the analysis of functions; in this analogy, the Riemann curvature tensor, of which the Ricci curvature is a natural by-product, would correspond to the full matrix of second derivatives of a function.
In mathematics, the Riemannian connection on a surface or Riemannian 2-manifold refers to several intrinsic geometric structures discovered by Tullio Levi-Civita, Élie Cartan and Hermann Weyl in the early part of the twentieth century: parallel transport, covariant derivative and connection form.