- PROPERTIES OF THE NORMALIZED RICCI FLOW 213
on a manifold of finite volume. To convert to the normalized fl.ow, define
dilating factors 1jJ ( t) > 0 so that the metrics g ( t) = 1jJ ( t) · g ( t) have constant
volume
r djj = 1,
}Mnand put t = J~ 1jJ ( T ) dT. Then dt / dt = 1jJ ( t), while the geometries of g and
g are related by the following lemma.
LEMMA 6.57. Let (Mn, g) be a Riemannian manifold. If g = 'I/Jg for
some 1jJ > 0, then the following relations result.
(1) The Levi-Civita connections of g and g are related by rt = I'fj.
(2) The - g (3, 1)f. -Riemann curvature tensors of g and g are related by
Rijk = Rijk·
(3) The (4, 0) -Riemann curvature tensors of g and g are related by
Rijkf. = 1/JRijk.f.·
( 4) The Ricci curvature tensors of g and g are related by Rij = Rj.(5) The scalar curvatures of g and g are related by R = 1/;-^1 R.
(6) The volume elements of g and g are related by djj = 1/Jn/^2 dμ.
Now writing equation 6.5 in the formwe see that
a a
- log det g = g^21 - (g· ·) = -2R
at at iJ '!!: r dμ = r Rdμ,
dt }Mn }Mn
hence that 1jJ is a smooth function of time. Thus we get the evolution
equation
(6.57) atg a = dt dtat a ('I/Jg)= -^2 - Rc+ ( 1/;^1 2 dt d'l/J) g.
Denote the average scalar curvature of g ( t) by
p ( t) ~ J Mn Rd_μ - =^1 R djj.
JMn dμ Mn
Then observing as above that
a 1 i. a n d'l(J
at logdet (1/;g) = -;j;g^1 at (1/Jgij) = -2R +-;{;di'we obtain0 = :t j djj = j :t log jdet ('I/Jg) djj
J (
= - 1/;R+ - --n a'ljJ) dμ _
21/J at
_ n d'ljJ
= - 1/Jp+--.
21/; dt