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 the second, induced, volume and/or form:
“It was not till the middle of the second dance, when, from some pauses in the movement wherein they all seemed to look up, I fancied I could distinguish an elevation of spirit different from that which is the cause or the effect of simple jollity.—In a word, I thought I beheld Religion mixing in the dance.”
—Laurence Sterne (1713–1768)
“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)
“I dare say I am compelled, unconsciously compelled, now to write volume after volume, as in past years I was compelled to go to sea, voyage after voyage. Leaves must follow upon each other as leagues used to follow in the days gone by, on and on to the appointed end, which, being Truth itself, is One—one for all men and for all occupations.”
—Joseph Conrad (1857–1924)
“The planter, the farmer, the mechanic, and the laborer ... form the great body of the people of the United States, they are the bone and sinew of the country—men who love liberty and desire nothing but equal rights and equal laws.”
—Andrew Jackson (1767–1845)