Search results
Results from the WOW.Com Content Network
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
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 ...
In mathematics, the measurable Riemann mapping theorem is a theorem proved in 1960 by Lars Ahlfors and Lipman Bers in complex analysis and geometric function theory. Contrary to its name, it is not a direct generalization of the Riemann mapping theorem , but instead a result concerning quasiconformal mappings and solutions of the Beltrami ...
A simple example of a statistical manifold, taken from physics, would be the canonical ensemble: it is a one-dimensional manifold, with the temperature T serving as the coordinate on the manifold. For any fixed temperature T , one has a probability space: so, for a gas of atoms, it would be the probability distribution of the velocities of the ...
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.
A Riemannian manifold is a smooth manifold together with a Riemannian metric. The techniques of differential and integral calculus are used to pull geometric data out of the Riemannian metric. For example, integration leads to the Riemannian distance function, whereas differentiation is used to define curvature and parallel transport.
The existence of isothermal coordinates can be proved by other methods, for example using the general theory of the Beltrami equation, as in Ahlfors (2006), or by direct elementary methods, as in Chern (1955) and Jost (2006). From this correspondence with compact Riemann surfaces, a classification of closed orientable Riemannian 2-manifolds ...
The relevant kernel is the standard Cauchy kernel (see Gakhov (2001); also cf. the scalar example below). An essential extension of the nonlinear method of stationary phase has been the introduction of the so-called finite gap g-function transformation by Deift, Venakides & Zhou (1997) , which has been crucial in most applications.