The First Induced Volume Form
Let Ω : Ψ(Rn+1)n+1 → R be a volume form defined on Rn+1. We can induce a volume form on M given by ω : Ψ(M)n → R given by ω(X1,…,Xn) := Ω(X1,…,Xn,ξ). This is a natural definition: in Euclidean differential geometry where ξ is the Euclidean unit normal then the standard Euclidean volume spanned by X1,…,Xn is always equal to ω(X1,…,Xn). Notice that ω depends on the choice of transverse vector field ξ.
Read more about this topic: Affine Differential Geometry
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