DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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Riemannian volume density

Definition

The Riemannian volume density is $μ_{g} = \sqrt{\det G_{x}} ∣ d x^{1} \dots d x^{n} ∣$ in coordinates.

The matrix is that of Coordinate criterion for a riemannian metric, so its determinant is positive and smooth. The density frames and their absolute-Jacobian law are Density bundle and smooth density fields. In dimension zero take the empty determinant to be one, giving weight one at every point, independently of orientation. The compatibility of these local formulas is proved in The riemannian volume density is coordinate independent ↗.

Source locator

Lee, Propositions 15.29–15.33 and Corollary 15.34, pp.389–391; density construction and integration pp.428–433.

Depends on

Used by

A conformal metric on the plane Example
The hyperbolic upper half space metric Example
Volume density in polar coordinates Example
The riemannian volume density is coordinate independent Lemma
Levi civita parallel transport preserves lengths angles and volume Proposition

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John M. Lee, Introduction to Smooth Manifolds, second edition (standard reference, not scraped)