2.8 R e m a r k s . (1) The formula (2-9) is closely related to the formula of theorem 5.5
in the paper [2] of Brothers.
(2) It is possible that M and gN do not have nonempty transverse intersection for any
g E G (in which case the set of g £ G with M 0 gN ^ 0 has measure zero and so (2-9)
reduces to 0 0). As an example of this let G be the additive group R2 and let M and N be
segments parallel to the cc-axis, say M = {(#,2/o) ' a x b}, N = {(z2/i) : c x d}.
In this case it is still possible to give a version of (2-9) which gives a nonzero result. This is
done by using the generalized coarea formula given in section 10 of the paper of Brothers
just quoted in the proof of (2-9) at the places where we use the coarea formula. For details
of this type of construction see section 11 of Brothers' paper and remark 3.10(2) below.
2.9 In proving the basic integral formula it can be assumed that M and N are embedded
submanifolds of G. To see this use a partition of unity on M x N to restrict the support of
h down to a subset of M x N of the form U x V where U is an open orientable submanifold
of M with smooth boundary, V is an open orientable submanifold of N both U and V are
embedded in G. Then prove (2-9) with M replaced by U and N replaced by V and then
sum over the partition of unity.
For the rest of this section we will use the following notation / : M x N G is the
(2-11) /«,»?) = ^ _ 1
Then for all g e G
= {(tv) &MxN: /({,,) = ^ = «?}•
By the coarea formula (see the appendix for the statement of this formula and for the
definition of the Jacobian «//(£, 77)),
(2-12) / / htlf-xMSl0(g)= [[ h(t,T,)Jf(t,T,)nMxN{t,T,).
What we will do is compute the Jacobian Jf((,7]) in terms of the geometric data (which
will relate its value to the angle a(T-LM^T±Nv)) and show that for almost all g £ G the
map (pg is a diffeomorphism of M D gN with f~1[g] and use this to relate the integrals
Sf-i[g) h^f-l[g] t o t n e integrals JMngN ho(pg QMngN-
In what follows we will use the standard isomorphism of T(MxN)^^ with TM^®TNV.
Vectors in T{M x iV)u7?) will be written as {X,Y) with X TMi and Y TM^.
2.10 Lemma. If{X,Y) G T(M x N){M then
(2-13) = RV-^(X - L^-UY)
Proof. It is enough to show f^^(X,0) = Rv-i*X and /*(^fi,)(0,y) = -Rq-i+Lfr-i+Y.
To show the first of these let c be a smooth curve in M with c'(0) = X. Then f+^^X =
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