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 ^