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While not derived as a Riemann sum, taking the average of the left and right Riemann sums is the trapezoidal rule and gives a trapezoidal sum. It is one of the simplest of a very general way of approximating integrals using weighted averages. This is followed in complexity by Simpson's rule and Newton–Cotes formulas.
This is a list of formulas encountered in Riemannian geometry. Einstein notation is used throughout this article. This article uses the "analyst's" sign convention for Laplacians, except when noted otherwise.
A Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as deformed versions of the complex plane : locally near every point they look like patches of the complex plane, but the global topology can be quite different.
Gromov's Betti number theorem. There is a constant C = C(n) such that if M is a compact connected n-dimensional Riemannian manifold with positive sectional curvature then the sum of its Betti numbers is at most C. Grove–Petersen's finiteness theorem.
The trapezoidal rule may be viewed as the result obtained by averaging the left and right Riemann sums, and is sometimes defined this way. The integral can be even better approximated by partitioning the integration interval, applying the trapezoidal rule to each subinterval, and summing the results. In practice, this "chained" (or "composite ...
The fundamental theorem asserts both existence and uniqueness of a certain connection, which is called the Levi-Civita connection or (pseudo-) Riemannian connection. However, the existence result is extremely direct, as the connection in question may be explicitly defined by either the second Christoffel identity or Koszul formula as obtained ...
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.
By the uniformization theorem X is either the Riemann sphere, the complex plane or the unit disk/upper halfplane. The first important invariant of a compact Riemann surface is its genus, a topological invariant given by half the rank of the Abelian group Γ/[Γ, Γ] (which can be identified with the homology group H 1 (X, Z)). The genus is zero ...
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