Differential geometry MOC

Volume form

Let $𝑀$ be an $𝑛$ -dimensional $𝐶 𝛼$ -manifold. A volume form $đ 𝑉 \in Ω 𝑛 (𝑀)$ is simply a choice of a top form, #m/def/geo/diff i.e. a differential form of rank $𝑛$ . In certain cases there exists a canonical volume form

Riemannian volume form for an oriented semi-Riemannian manifold.

Properties

A nonvanishing volume form $đ 𝑉$ induces an orientation on $𝑀$ .
$𝑀$ is orientable iff it admits a nonvanishing volume form $đ 𝑉$ .

Proof

#missing/proof

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