The Second Induced Volume Form
For tangent vectors X1,…,Xn let H := (hi,j) be the n × n matrix given by hi,j := h(Xi,Xj). We define a second volume form on M given by ν : Ψ(M)n → R, where ν(X1,…,Xn) := |det(H)|½. Again, this is a natural definition to make. If M = Rn and h is the Euclidean scaler product then ν(X1,…,Xn) is always the standard Euclidean volume spanned by the vectors X1,…,Xn. Since h depends on the choice of transverse vector field ξ it follows that ν does too.
Read more about this topic: Affine Differential Geometry
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