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
Famous quotes containing the words induced, volume and/or form:
“It is a misfortune that necessity has induced men to accord greater license to this formidable engine, in order to obtain liberty, than can be borne with less important objects in view; for the press, like fire, is an excellent servant, but a terrible master.”
—James Fenimore Cooper (1789–1851)
“A big leather-bound volume makes an ideal razorstrap. A thin book is useful to stick under a table with a broken caster to steady it. A large, flat atlas can be used to cover a window with a broken pane. And a thick, old-fashioned heavy book with a clasp is the finest thing in the world to throw at a noisy cat.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)
“The sense of an entailed disadvantage—the deformed foot doubtfully hidden by the shoe, makes a restlessly active spiritual yeast, and easily turns a self-centred, unloving nature into an Ishmaelite. But in the rarer sort, who presently see their own frustrated claim as one among a myriad, the inexorable sorrow takes the form of fellowship and makes the imagination tender.”
—George Eliot [Mary Ann (or Marian)