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:
“Few can be induced to labor exclusively for posterity; and none will do it enthusiastically. Posterity has done nothing for us; and theorize on it as we may, practically we shall do very little for it, unless we are made to think we are at the same time doing something for ourselves.”
—Abraham Lincoln (1809–1865)
“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)
“To do the opposite of something is also a form of imitation, namely an imitation of its opposite.”
—G.C. (Georg Christoph)