Riemannian manifold

== English == === Etymology === Named after German mathematician Bernhard Riemann (1826–1866). See also Riemannian. === Noun === Riemannian manifold (plural Riemannian manifolds) (differential geometry, Riemannian geometry) A real, smooth differentiable manifold whose each point has a tangent space equipped with a positive-definite inner product;(more formally) an ordered pair (M, g), where M is a real, smooth differentiable manifold and g its Riemannian metric. ==== Usage notes ==== Not to be confused with Riemann surface. Riemannian manifolds are the principal subject of study in Riemannian geometry. Formally, a Riemannian manifold is defined as the ordered pair ( M , g ) {\displaystyle (M,g)} of the manifold and the Riemannian metric with which it is equipped. Except in very formal contexts, however, the term is used as if referring to a type of manifold. The Riemannian metric, a tensor, is also said to be smooth, but in a technically different sense as when used for the manifold. ==== Synonyms ==== Riemannian space ==== Hypernyms ==== differentiable manifold, smooth manifold ==== Derived terms ==== pseudo-Riemannian manifold semi-Riemannian manifold ==== Translations ==== === See also === differentiable manifold Riemannian metric (= Riemannian metric tensor) === Further reading === Riemannian geometry on Wikipedia.Wikipedia Differentiable manifold on Wikipedia.Wikipedia Tangent space on Wikipedia.Wikipedia Metric tensor on Wikipedia.Wikipedia