Riemannian manifold

English Etymology Named after German mathematician Bernhard Riemann (1826–1866). See also en Riemannian. Noun en differential geometry Riemannian geometry A real, smooth differentiable manifold whose each point has a tangent space equipped with a positive-definite inner product;en more formally an ordered pair (M, g), where M is a real, smooth differentiable manifold and g its Riemannian metric. en By definition, a Riemannian manifold has at each point a tangent space equipped with a positive-definite inner product, ; information about these inner products is encoded in the Riemannian metric tensor, . en 1984 Isaac Chavel Eigenvalues in Riemannian Geometry https://books.google.com.au/books?id=0v1VfTWuKGgC&pg=PA55&dq=%22Riemannian+manifold%22%7C%22Riemannian+manifolds%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwjRyoyb2fvrAhVwzTgGHQlgBGIQ6AEwTHoECCAQAg#v=onepage&q=%22Riemannian%20manifold%22%7C%22Riemannian%20manifolds%22&f=false 55 Academic Press In this chapter we extend the study of eigenvalues to Riemannian manifolds whose curvature may not be constant, but is, nevertheless, bounded. en 2000 Daniel W. Stroock An Introduction to the Analysis of Paths on a Riemannian Manifold w:American Mathematical Society https://books.google.com.au/books?id=Ugy9AwAAQBAJ&pg=PA165&dq=%22Riemannian+manifold%22%7C%22Riemannian+manifolds%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwjRyoyb2fvrAhVwzTgGHQlgBGIQ6AEwD3oECEQQAg#v=onepage&q=%22Riemannian%20manifold%22%7C%22Riemannian%20manifolds%22&f=false 165 Further, a much harder theorem, due to J. Nash [31], says that a separable Riemannian manifold of dimension can be isometrically embedded in with . en 2018 John M. Lee Introduction to Riemannian Manifolds 2nd https://books.google.com.au/books?id=VUOCDwAAQBAJ&pg=PA55&dq=%22Riemannian+manifold%22%7C%22Riemannian+manifolds%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwjRyoyb2fvrAhVwzTgGHQlgBGIQ6AEwCHoECDUQAg#v=onepage&q=%22Riemannian%20manifold%22%7C%22Riemannian%20manifolds%22&f=false 55 Springer Before we delve into the general theory of Riemannian manifolds, we pause to give it some substance by introducing a variety of "model Riemannian manifolds" that should help to motivate the general theory. Usage notes Not to be confused with en Riemann surface. Riemannian manifolds are the principal subject of study in Riemannian geometry. Formally, a Riemannian manifold is defined as the ordered pair 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 en Riemannian space Hypernyms en differentiable manifold, en smooth manifold Derived terms en pseudo-Riemannian manifold en semi-Riemannian manifold Translations real, smooth differentiable manifold equipped with a Riemannian metric Bulgarian: bg риманово многообразие n Finnish: fi Riemannin monisto French: fr variété riemannienne f German: de riemannsche Mannigfaltigkeit f, de riemannscher Raum m Italian: it varietà riemanniana f Polish: pl rozmaitość riemannowska f, pl przestrzeń Riemanna f Russian: ru риманово многогобразие n ри́маново многообра́зие See also en differentiable manifold en Riemannian metric (= en Riemannian metric tensor) Further reading Riemannian geometry Differentiable manifold Tangent space Metric tensor en eponyms en Geometry Manifolds