Local Orthonormal Coframe defines Riemannian Density

Theorem

Let $\struct {M, g}$ be an $n$-dimensional Riemannian manifold.

Let $\tuple {\epsilon^i}$ be an local orthonormal coframe.

Let $\mu$ be a Riemannian density.

Then there exists $\mu$ such that:

$\mu = \size {\epsilon^1 \wedge \ldots \wedge \epsilon^n}$

where $\wedge$ denotes the wedge product.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Basic Constructions on Riemannian Manifolds