- THE RICCI FLOW OF HOMOGENEOUS GEOMETRIES 9
metric g.. Then one may define a vector space isomorphism g --+ /\^2 g by
Composing this with the commutator, regarded as the linear map /\^2 g --+ g
sending V /\ W f---7 [V, W], yields a vector space endomorphism r : g --+ g
whose matrix with respect to the ordered basis (3 = (F 1 , F 2 , F 3 ) is
Observe that the matrix r f3 displays all the structure constants of g.
Suppose further that g is unimodular. In this case, one has tr ad V = 0
for every V E g, which implies in particular that r is self adjoint with respect
tog. So by an orthogonal change of basis (3 f---7 o: = (P 1 ,F 2 ,P 3 ), the matrix
r f3 of r can be transformed into
(1.1)
where the c~j are the Lie algebra structure constants for {Pi}. We can
arrange that >., μ, v E { -1, 0, 1} without changing the metric g if we are
willing to let the frame J Fi} be merely orthogonal. For instance, if >., μ, v
are all nonzero, we get >., P,, i/ = ±1 by putting
The general case is just as easy. We can also arrange that >. ::::; μ ::::; v.
For instance, if F1 = -F2, F2 = F1, F3 = F3, the orientation of the
manifold is unchanged, but (5., P,, v) = (μ, >., v). Finally if we are willing
to reverse the orientation of g, we can arrange that there are at least as
many negative structure constants as positive, thereby displaying exactly six
distinct signatures. This construction and classification was first described
in [98], so we call an orthogonal frame field for which>.::::;μ::::; v E {-1, 0, 1}
a Milnor frame for the left-invariant metric g. Table 1 (above) displays
the possible simply-connected unimodular Lie groups.
Let {Fi} be a Milnor frame for some left-invariant metric g on a 3-
dimensional unimodular Lie group g. Then there are positive constants A,
B, and C so that g may be written with respect to the set { wi} of 1-forms
dual to {Fi} as
(1.2) g = Aw^1 @w^1 + Bw^2 @w^2 + Cw^3 @w^3.