Riemannian manifold

English

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Riemannian manifold

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Etymology

Named after German mathematician Bernhard Riemann (1826–1866). See also Riemannian.

Noun

Riemannian manifold (plural Riemannian manifolds)

1. (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.

   By definition, a Riemannian manifold ${\displaystyle M}$ has at each point ${\displaystyle p}$ a tangent space ${\displaystyle T_{p}M}$ equipped with a positive-definite inner product, ${\displaystyle g_{p}}$; information about these inner products is encoded in the Riemannian metric tensor, ${\displaystyle g}$.

   • 1984, Isaac Chavel, Eigenvalues in Riemannian Geometry, Academic Press, page 55:

     In this chapter we extend the study of eigenvalues to Riemannian manifolds whose curvature may not be constant, but is, nevertheless, bounded.

   • 2000, Daniel W. Stroock, An Introduction to the Analysis of Paths on a Riemannian Manifold, American Mathematical Society, page 165:

     Further, a much harder theorem, due to J. Nash [31], says that a separable Riemannian manifold of dimension ${\displaystyle d}$ can be isometrically embedded in ${\displaystyle \textstyle \mathbb {R} ^{N}}$ with ${\displaystyle N\leq {\tfrac {1}{2}}d(d+1)(3d+11)}$.

   • 2018, John M. Lee, Introduction to Riemannian Manifolds, 2nd edition, Springer, page 55:

     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 Riemann surface.
• Riemannian manifolds are the principal subject of study in Riemannian geometry.
• Formally, a Riemannian manifold is defined as the ordered pair ${\displaystyle (M,g)}$ 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