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)
“F.R. Leavis’s “eat up your broccoli” approach to fiction emphasises this junkfood/wholefood dichotomy. If reading a novel—for the eighteenth century reader, the most frivolous of diversions—did not, by the middle of the twentieth century, make you a better person in some way, then you might as well flush the offending volume down the toilet, which was by far the best place for the undigested excreta of dubious nourishment.”
—Angela Carter (1940–1992)
“I don’t think of form as a kind of architecture. The architecture is the result of the forming. It is the kinesthetic and visual sense of position and wholeness that puts the thing into the realm of art.”
—Roy Lichtenstein (b. 1923)