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
Famous quotes containing the word conclusion:
“The source of our actions resides in an unconscious propensity to regard ourselves as the center, the cause, and the conclusion of time. Our reflexes and our pride transform into a planet the parcel of flesh and consciousness we are.”
—E.M. Cioran (b. 1911)
“A certain kind of rich man afflicted with the symptoms of moral dandyism sooner or later comes to the conclusion that it isn’t enough merely to make money. He feels obliged to hold views, to espouse causes and elect Presidents, to explain to a trembling world how and why the world went wrong. The spectacle is nearly always comic.”
—Lewis H. Lapham (b. 1935)