Riemannian manifold en · NOUN
Etymology
Named after German mathematician Bernhard Riemann (1826–1866). See also Riemannian.
Meanings
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(qualifier:differential geometry; Riemannian geometry; differential geometry; Riemannian geometry) A real, smooth differentiable manifold whose each point has a tangent space equipped with a positive-definite inner product;
In this chapter we extend the study of eigenvalues to Riemannian manifolds whose curvature may not be constant, but is, nevertheless, bounded.
1984, Isaac Chavel, Eigenvalues in Riemannian Geometry, Academic Press, page 55:By definition, a Riemannian manifold M has at each point p a tangent space T#95;pM equipped with a positive-definite inner product, g#95;p; information about these inner products is encoded in the Riemannian metric tensor, g.
Further, a much harder theorem, due to J. Nash [31], says that a separable Riemannian manifold of dimension d can be isometrically embedded in #92;textstyle#92;mathbbRᴺ with N#92;le#123;#92;tfrac 1 2#125;d(d#43;1)(3d#43;11).
2000, Daniel W. Stroock, An Introduction to the Analysis of Paths on a Riemannian Manifold, American Mathematical Society, page 165: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.
2018, John M. Lee, Introduction to Riemannian Manifolds, 2nd edition, Springer, page 55:
- (qualifier:differential geometry; Riemannian geometry; more formally) 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. ▸ an ordered pair (M, g), where M is a real, smooth differentiable manifold and g its Riemannian metric.
Forms
| Spelling | Features | Labels | Source |
|---|---|---|---|
| Riemannian manifolds | Number=Plur | lexicographic |
Deriveds
semi-Riemannian manifold · pseudo-Riemannian manifold
Hypernyms
smooth manifold · differentiable manifold
Relateds
Riemannian metric · differentiable manifold
Synonyms
Translations (9)
fr variété riemannienne (real, smooth differentiable manifold equipped with a Riemannian metric) · pl rozmaitość riemannowska (real, smooth differentiable manifold equipped with a Riemannian metric) · de riemannscher Raum (real, smooth differentiable manifold equipped with a Riemannian metric) · pl przestrzeń Riemanna (real, smooth differentiable manifold equipped with a Riemannian metric) · fi Riemannin monisto (real, smooth differentiable manifold equipped with a Riemannian metric) · ru ри́маново многообра́зие (real, smooth differentiable manifold equipped with a Riemannian metric) · it varietà riemanniana (real, smooth differentiable manifold equipped with a Riemannian metric) · de riemannsche Mannigfaltigkeit (real, smooth differentiable manifold equipped with a Riemannian metric) · bg риманово многообразие (real, smooth differentiable manifold equipped with a Riemannian metric)