Math 537B Homework 4 (Optional)
April 24, 2006
1) Recall that in Homework 3 we derived that the metric in normal coordinates can be expanded
as a Taylor series as
gij (x) = ij 1
3Rikj` (0) xkx`+Ojxj3:
Use this and the well-known formulas
d
dt log det g=gij d
dtgij
d
dtgij =gik d
dtgk`g`j
to show that the Taylor series for det gis
det g(x) = 1 1
3Rij (0) xixj+Ojxj3
and that the Taylor series for the volume of the ball of radius r, called V(r);is
V(r) = 11
6 (n+ 2)S(0) r2+Or3rn!n1
n
where !n1is the (n1)-dimensional volume of the sphere of radius 1 in Rnand Sis the scalar
curvature. Note that rn!n1=n is equal to the volume of the Euclidean ball of radius r:
2) Recall that the Lie derivative of a covariant tensor Tis de…ned as
LXT= lim
t!0
tTT
t
where tis a 1-parameter family of di¤eomorphisms t:M!Mgenerated by the vector …eld X:
Show that if gij is the Riemannian metric tensor, then
(LXg)ij =riXj+rjXi
where ris the Riemannian connection and Xj=gjk Xkif X=Xk@
@xk:Conclude that if gij (t) =
tgij (0) is a solution to the normalized Ricci ‡ow, then
(rR)gij =riXj+rjXi:
A Riemannian metric satisfying this equation (in two dimensions) for some vector …eld Xis called a
Ricci-soliton. The Ricci soliton equation in higher dimensions is
rgij Rij =riXj+rjXi:
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