Riemannian manifold

Etymology
Named after German mathematician (1826–1866). See also.

Noun

 * 1)  A real, smooth differentiable manifold whose each point has a tangent space equipped with a positive-definite inner product;  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.
 * Riemannian manifolds are the principal subject of study in Riemannian geometry.
 * Formally, a Riemannian manifold is defined as the ordered pair $$(M,g)$$ 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.

Translations

 * French: variété riemannienne
 * German: riemannsche Mannigfaltigkeit, riemannscher Raum
 * Italian: varietà riemanniana
 * Russian: риманово многогобразие