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
Famous quotes containing the words induced, volume and/or form:
“The classicist, and the naturalist who has much in common with him, refuse to see in the highest works of art anything but the exercise of judgement, sensibility, and skill. The romanticist cannot be satisfied with such a normal standard; for him art is essentially irrational—an experience beyond normality, sometimes destructive of normality, and at the very least evocative of that state of wonder which is the state of mind induced by the immediately inexplicable.”
—Sir Herbert Read (1893–1968)
“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)
“‘A thing is called by a certain name because it instantiates a certain universal’ is obviously circular when particularized, but it looks imposing when left in this general form. And it looks imposing in this general form largely because of the inveterate philosophical habit of treating the shadows cast by words and sentences as if they were separately identifiable. Universals, like facts and propositions, are such shadows.”
—David Pears (b. 1921)