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In complex analysis, the Riemann mapping theorem states that if is a non-empty simply connected open subset of the complex number plane which is not all of , then there exists a biholomorphic mapping (i.e. a bijective holomorphic mapping whose inverse is also holomorphic) from onto the open unit disk
exponential map (Riemannian geometry) for a manifold with a Riemannian metric, exponential map (Lie theory) from a Lie algebra to a Lie group, More generally, in a manifold with an affine connection, (), where is a geodesic with initial velocity X, is sometimes also called the exponential map. The above two are special cases of this with ...
The exponential map of the Earth as viewed from the north pole is the polar azimuthal equidistant projection in cartography. In Riemannian geometry, an exponential map is a map from a subset of a tangent space T p M of a Riemannian manifold (or pseudo-Riemannian manifold) M to M itself. The (pseudo) Riemannian metric determines a canonical ...
A K-quasiregular map R n → R n can omit at most a finite set. When n = 2, this omitted set can contain at most one point (this is a simple extension of Picard's theorem). But when n > 2, the omitted set can contain more than one point, and its cardinality can be estimated from above in terms of n and K .
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 ...
The measurable Riemann mapping theorem shows more generally that the map to an open subset of the complex sphere in the uniformization theorem can be chosen to be a quasiconformal map with any given bounded measurable Beltrami coefficient.
Suppose that is a smooth, simple, closed contour in the complex plane. [2] Divide the plane into two parts denoted by + (the inside) and (the outside), determined by the index of the contour with respect to a point. The classical problem, considered in Riemann's PhD dissertation, was that of finding a function
If is compact, it has a Riemannian metric invariant under left and right translations, then the Lie-theoretic exponential map for coincides with the exponential map of this Riemannian metric. For a general G {\displaystyle G} , there will not exist a Riemannian metric invariant under both left and right translations.