Differential geometry MOC

Oriented manifold

Let (𝑀,𝒜) be a 𝐶𝛼-manifold. An atlas ℬ ⊆𝒜 is called oriented iff all transition maps have positive Jacobian determinant. An orientation on 𝑀 is the choice of a maximal oriented atlas. #m/def/geo/diff If an orientation exists we say that 𝑀 is orientable.

• If 𝑀 is semi-Riemannian, we can use orientation to define a Riemannian volume form as well as the Hodge star.

Properties

• A manifold is orientable iff there exists a nonvanishing volume form

See also

• Oriented vector space

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