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.
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