Riemannian metric

黎曼度量

Geometry & Topology · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

A smooth symmetric covariant two-tensor on a smooth real manifold whose value at every point is a positive definite bilinear form on that point's tangent space.

光滑实流形上的光滑对称二次协变张量，在每一点的切空间上给出正定双线性型。

Example

The Euclidean metric $dx^2+dy^2$ becomes $dr^2+r^2d\theta^2$ in polar coordinates for $r>0$; a vector $a\partial_r+b\partial_\theta$ has squared length $a^2+r^2b^2$.

Usage and distinctions

The tensor measures tangent vectors. The induced distance between points is obtained by infima of curve lengths; an arbitrary metric-space distance is not such a tensor.

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Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Liviu I. Nicolaescu — Lectures on the Geometry of Manifolds§2.3.2, Definition 2.3.4(c); §4.1.1, Definitions and examples · Checked 2026-10-14

Translation sources

1. Zhang Sirong — Beihang University, 微分流形与黎曼几何, Chapter 3 slidesChapter 3，黎曼度量，Definition on slide 47/54; PDF p. 12 · Checked 2026-10-14

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