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Riemannian manifolds are named after German mathematician Bernhard Riemann, who first conceptualized them. Formally, a Riemannian metric (or just a metric) on a smooth manifold is a choice of inner product for each tangent space of the manifold. A Riemannian manifold is a smooth manifold together with a Riemannian metric.
Pages in category "Riemannian manifolds" The following 41 pages are in this category, out of 41 total. This list may not reflect recent changes. * Riemannian manifold; A.
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as smooth manifolds with a Riemannian metric (an inner product on the tangent space at each point that varies smoothly from point to point). This gives, in particular, local notions of angle, length of curves, surface area and volume.
In higher dimensions, a manifold may have different curvatures in different directions, described by the Riemann curvature tensor. In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point.
In mathematics, a Riemannian manifold is said to be flat if its Riemann curvature tensor is everywhere zero. Intuitively, a flat manifold is one that "locally looks like" Euclidean space in terms of distances and angles, e.g. the interior angles of a triangle add up to 180°. The universal cover of a complete flat manifold
Any oriented pseudo-Riemannian (including Riemannian) manifold has a natural volume form. In local coordinates , it can be expressed as ω = | g | d x 1 ∧ ⋯ ∧ d x n {\displaystyle \omega ={\sqrt {|g|}}dx^{1}\wedge \dots \wedge dx^{n}} where the d x i {\displaystyle dx^{i}} are 1-forms that form a positively oriented basis for the ...
Let (,) be a complete and smooth Riemannian manifold of dimension n. If k is a positive number with Ric g ≥ (n-1)k, and if there exists p and q in M with d g (p,q) = π/ √ k, then (M,g) is simply-connected and has constant sectional curvature k.
A Riemannian manifold is a differentiable manifold in which each tangent space is equipped with an inner product , in a manner which varies smoothly from point to point. Given two tangent vectors u {\displaystyle u} and v {\displaystyle v} , the inner product u , v {\displaystyle \langle u,v\rangle } gives a real number.