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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 ...
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 .
The definition of a residue can be generalized to arbitrary Riemann surfaces. Suppose ω {\displaystyle \omega } is a 1-form on a Riemann surface. Let ω {\displaystyle \omega } be meromorphic at some point x {\displaystyle x} , so that we may write ω {\displaystyle \omega } in local coordinates as f ( z ) d z {\displaystyle f(z)\;dz} .
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 ...
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.
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold (i.e., it is a tensor field).
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 ...
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.