Riemannian manifoldとは 意味・読み方・使い方

名詞

Riemannian manifold (複数形 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, Springer, 2nd Edition, 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.

使用する際の注意点

• 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.

同意語

• Riemannian space

上位語

• differentiable manifold, smooth manifold

派生語

• pseudo-Riemannian manifold
• semi-Riemannian manifold

参考

• differentiable manifold
• Riemannian metric (= Riemannian metric tensor)

Further reading

• Riemannian geometry on Wikipedia.
• Differentiable manifold on Wikipedia.
• Tangent space on Wikipedia.
• Metric tensor on Wikipedia.