THE BASIC INTEGRAL FORMULA 9

•^l\t=oc(t)rj~1 = R^-i+X. To show the second recall that if c and c\ are curves in a Lie

group then j-t(c(i)ci(t)) — RCl(t)*c'{t) + £

c

(t)* c i(0 (f°r example this follows from the

"Leibnitz formula" on page 14 of [14] Vol. 1). If ci(t) = c(2)

- 1

then c(t)ci(t) is constant

whence 0 = Rc{t)-i*c'(t) + Lc{iftc{t)-1 i.e. ^ c ( t ) " 1 = -Lc{t)-i*Rc{t)-i+c'{t). Now let

c be a smooth curve in N with c'(0) = Y. Then using what was just shown and that left

and right translation commute,

w°^)=|

= Lt*dt

d

city1

\t=0

= —L^LJJ-I+RTJ-I+Y

= —Rv-i*L^ri-i^Y

This completes the proof.

2.11 Lemma. The kernel of f^^ is {(X,L^-i^X) : X G TM^ fl L^-i^TN^} and

the image of f(zlV)* is R^-i^TM^ -f L^-x + TN^). Therefore (£,77) is a regular point of f

if and only ifTM^ + L^-^TN^ = TG^.

Proof. See the appendix for the definition of a regular point. This lemma follows directly

from the last one.

2.12 Lemma. For all g G G define a function ng : f~l[g\ —• M fl gN by

(2-H) *„{*,*) = *•

Then for all g G G, ipg is a bijection of M C\ gN onto /-1[7] and the inverse of (pg is

7vg. If g is a regular value of f then M and gN intersect transversely and thus M fl gN

is a smooth submanifold of G for almost all g G G. If g is a regular value of f then

tp9 : M fl gN — » /-1[7] is a diffeomorphism.

Proof. That (pg is a bijection with inverse 7rg is left to the reader. If g is a regular value

of / and £ G M 0 gN then let rj G N with £ — gr\. Thus g = £?7_1 = f(^,rj) and as g is a

regular value of / using lemma 2.11 in the last line,

TM$ + T(gN)t = TMt + Lg«TNv

= TGt.

This proves M and gN intersect transversely when g is a regular value of / , and by Sard's

theorem (see appendix) almost every g G G is a regular value of / .

If g is a regular value of / then f~1[g] is an embedded submanifold of M x N and

M DgN is a submanifold of G as M and gN intersect transversely. From the definitions of

(pg and -Kg it is clear they are both smooth functions and as they are inverse to each other

this implies that both are diffeomorphisms. This completes the proof.