NOTES AND COMMENTARY 19where a and /3 are constant and 'Y is a linear function of e such that ( ~
e-°' ( d-y /dB) > 0. It is not hard to check that a Milnor frame for g is given
by
F1 = e--y/^2 !!__ -e'Y/^2 !!__
ax ay
4 -a a
F2 = ( e aeF3 = e--y/^2 !!__ + e'Y/^2 !!__.
ax ayIndeed, one has [F1, F2] = 2F3, [F2, F3] = -2F1, [F3, F1] = 0, and can write
the metric ( 1.12) in the form
16
g = 2ef3 w^1 ® w^1 + ( 2 w^2 ® w^2 + 2ef3 w^3 ® w^3.Clearly, g descends to a metric on the product of the line and the torus 72 ,
and A acts on IR x 72 by (B, x, y) I--+ (e + 2Jr, >._x, >-+y). If
w ( e + 27r) = w ( e) + 2 log>-+,
then A is an isometry, whence g becomes a well defined metric on the com-
pact mapping torus NX, regarded as a twisted 72 bundle over 51. By (1.12),
it is easy to see that a governs the length of the base circle, while f3 and 'Y
describe the scale and skew of the fibers, respectively. Under the Ricci flow,
the metric (1.12) evolves as follows: f3 and 'Y remain fixed, while one has
a (t) =a (0) + ln Ji+ (2t.
Intuitively, one may interpret this by saying that the Ricci flow attempts to
'untwist' the bundle NX_.Notes and commentary
Some good sources for basic 3-manifold topology are [71], [78], and
[70]. Homogeneous Riemannian spaces are reviewed in Chapter 3 of [27]
and Chapter 7 of [20]. The curvatures ofleft-invariant metrics on Lie groups
are studied in [98].
Detailed analyses of the behavior of the normalized and unnormalized
Ricci flows on the remaining homogeneous geometries may be found in [76]
and [86], respectively. Qualitatively, one sees a variety of behaviors. For
example, metrics in the Is~^2 ) family converge. Metrics in the 7-l^3 family
converge under the normalized Ricci flow, and give rise to immortal solu-
tions of the unnormalized flow. (See Section 4 of Chapter 2.) Under the
normalized flow, compact manifolds in the s1(2,JR) and 7-l^2 x IR families
collapse to 2-dimensional limits, while compact manifolds in the 52 x IR and
sol families collapse to 1-dimensional limits.