12 THE KINEMATIC FORMULA IN RIEMANNIAN HOMOGENEOUS SPACES
We still have to relate this to the angle ^{T^M^T^N^). To do this let
U = span{v
f c + 1
,..., v
p
,w
k + u
™J-
This is a vector space of dimension n k. Complete u j .
+ 1
, . . . ,v
p
to an orthonormal
basis 1;*.+1?..., vn of U. Then v
p
-fi,..., vn is an orthonormal basis of
V1-
L^-i^T^M^.
Likewise if tufc+i,... ,ttfg is completed to an orthonormal basis wk+i,... ,wn of U then
wq+i,... ,iu
n
is a basis of
W"1
=
Lv-i*T±Nv.
Because the dimension of U is n k the Hodge star on U maps
/\r(U)
to / \ ((7)
(see the book [11] page 15) and
vk+i A A vp = ± * (v
p + 1
A A vn)
Wk+I A A wq = ± * (wq+1 A A iyn)
Using known identities for * (see page 16 of [11]) and the last two equations,
vk+1 A A vpAwk+i A A wq
= ±{vk+1 A A vp) A *(wq+1 A A wn)
±(*vk+1 A A vp) A wq+i A A wn
= ±vp+1 A A vn A Wg+i A A iun.
Using this in (2-20) and recalling the definition of ^T^M^T^N^),
Jf{t,l) = 2* A(T7)||V
P +
I A A vn A wq+1 A A wn\\
(2-21) =
2A(71)r(T-LMt,T±Nri).
2.14 It remains to relate Jr-it^
h^lf-1[g]
to JMH
N^1
° Pg ^MngN- Let g be a regular
value of / . Then, by 2.12, pg : M fl gN »
/_1[7]
is a diffeomorphism with inverse 7Tg, If
(£,?/) 6 /
- 1
[# ] then, using the notation of equation (2-16), Z\,...,Zk is an orthonormal
basis of Kernel(/Jte(^7?)) =
T{f~~l[g])^^y
From the definition of irg and Z{ it is clear
TCg*Zi = 4j7r9*(Xt-, A\-) = -75-^i- But Xi,...,
Arfc
is an orthonormal basis of T(M 0 gN)%
and ^ is the inverse of (pg. Therefore we have just shown yg+Xi = v2Z{ for 1 i k.
This implies (f*Q,f-i^ =
2k/2
FlMngN, so that by the change of variable formula,
/ h$lf-ng]=2* / ho(pgttMngN.
Jf-l[g]
JMngN
Using this equation and equation (2-21) in equation (2-12) yields (2-9) and completes the
proof of the basic integral formula.
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