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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.
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 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 ...
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.
Objects appearing in the formulas of Gauss, such as the Christoffel symbols, can be given a natural geometric interpretation in this framework. Unlike the more intuitive normal bundle , easily visualised as a tubular neighbourhood of an embedded surface in E 3 , the frame bundle is an intrinsic invariant that can be defined independently of an ...
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space , the n {\displaystyle n} -sphere , hyperbolic space , and smooth surfaces in three-dimensional space, such as ellipsoids and paraboloids , are all examples of ...
In geometry, the Riemann sphere is the prototypical example of a Riemann surface, and is one of the simplest complex manifolds. In projective geometry , the sphere is an example of a complex projective space and can be thought of as the complex projective line P 1 ( C ) {\displaystyle \mathbf {P} ^{1}(\mathbf {C} )} , the projective space of ...
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