10 THE KINEMATIC FORMULA IN RIEMANNIAN HOMOGENEOUS SPACES
2.13 We now compute the Jacobian (Jf){^rj) at a regular point (Cv) °f /• First some
notation. Let (€,rj) be a regular point of / and set
n = dim((x), p = dim(M), q = dim(iV), k = dim(Kernel(/*(£|17)))
Then, using 2.11,
fc = p + g - n = dim(TM{ fl L^v-iTNv).
Let X i , . . . ,Xk be an orthonormal basis of TM{ fl L^v-i^TNv. Then, as the metric is
left invariant,
(2-15) Yi^Lfr-i+Xi lik
is an orthonormal basis of L^-i^TM^ fl TNV.
Complete X\,..., Xk to an orthonormal basis X\,..., Xp of TM{ and Y\,..., Yj. to an
orthonormal basis y
1 ?
. . . , Fg of TTV^. From 2.11 it follows that
(2-16) Zi = ~(Xi,Lvi-l,Xi)=-^=(Xi,Yi) lik
is an orthonormal basis of Kernel(/#(^)1?)) and therefore if
(2-17) Wi = ±{Xi,-Lv(-t.Xi) = +=(XU -Yi) lik
then the p + q n vectors
(2-18) Wlt...,Wk, (Xh+1,0),...,{Xp,0), ( 0 , n + , ) , . . . , (0,y,)
are an orthonormal basis of Kerne^/*^^))-1-. Using lemma 2.10
= —fcRri-i+Xi + Rv-i+L^-iLijt-i+Xi
= ^ 2 / ^ - 1 ^
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