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.