The Conclusion
It can be shown that there is, up to sign, a unique choice of transverse vector field ξ for which the two conditions that ∇ω ≡ 0 and ω ≡ ν are both satisfied. These two special transverse vector fields are called affine normal vector fields, or sometimes called Blaschke normal fields. From its dependence on volume forms for its definition we see that the affine normal vector field is invariant under volume preserving affine transformations. These transformations are given by SL(n+1,R) ⋉ Rn+1, where SL(n+1,R) denotes the special linear group of (n+1) × (n+1) matrices with real entries and determinant 1, and ⋉ denotes the semi-direct product. SL(n+1,R) ⋉ Rn+1 forms a Lie group.
Read more about this topic: Affine Differential Geometry
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