2. T H E BASI C I N T E G R A L FORMUL A
FOR SUBMANIFOLD S OF A LIE GROUP .
2.1 We start with a discussion on angles between subspaces. If V is an n dimensional
and W is an m dimensional subspace of an inner product space with inner product (,)
then let v i , . . . , vn k e a n orthonormal basis of V and tui,.. . ,wm an orthonormal basis of
W and define
(2-1) r(V, W) - llvj A A vn A
Wl
A A wm\\
where
(2-2) ||«i A...Aajfc||2 = det((aj ,aJi)).
If V and W are both one dimensional then cr(V,W) = \ sm$\ where 6 is the angle between
V and W. In general 0 a(V, W) 1 with r(V, W) = 0 if and only if V 0 W ^ .{0} and
r(F, W) 1 if and only if V is orthogonal to W. Also if /o is a linear isometry of the inner
product space containing V and W into some other inner product space then
r(pV,pW) = r{V,W)
(2-3) cr{V,W) = r{V,W)
2.2 Let G be a Lie group and £ £ (7. Then left and right translation by £ on (7 will be
denoted by L{ and i?^ respectively, that is L^(g) = £g and R^(g) = g£. Left translation
can be used to identify all tangent spaces to G with TGe, the tangent space to G at the
identity element e. Assume that G has a left invariant metric (,) then this identification
of the tangent spaces of G with each other allows the above definition of angles to be
extended to compare angles between subspaces of tangent spaces to G at different points.
To be exact if V is a subspace of TG% and W is a subspace of TGV then set
(2-4) a(V,W) = a(L(-imV,Ln-i,W).
With this definition it follows that for all g £ G
(2-5) r{LgtV,W) = *{V,Lg.W) = a(V,W).
Also if a £ G and the metric is invariant by Ra then
(2-6) r{RamV,Ra.W) i f « , } = (,)
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