Riemannian manifold

English

Etymology

Named after German mathematician Template:W (1826–1866). See also Template:M.

Noun

1. Template:Lb A real, smooth differentiable manifold whose each point has a tangent space equipped with a positive-definite inner product;
   Template:Lb an ordered pair (M, g), where M is a real, smooth differentiable manifold and g its Riemannian metric.

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Usage notes

• Not to be confused with Template:M.
• 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.

Synonyms

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Hypernyms

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Derived terms

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Translations

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See also

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• Template:L (= Template:L)

Further reading

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