2 1. RICCI SOLITONS
Some highlights of this chapter are a systematic description of the equa
tions obtained from differentiating the gradient Ricci soliton equation, the
constructions of the Bryant steady Ricci soliton and the rotationally sym
metric expanding Ricci soliton with positive curvature operator, examples of
homogeneous Ricci solitons, introduction to Perelman's energy and entropy
functional via gradient Ricci solitons, and the Buscher duality transforma
tion.
1. General solitons and their canonical forms
We begin by recalling the following.
DEFINITION
1.1 (General Ricci soliton). A solution g(t) of the Ricci flow
on
Mn
is a Ricci soliton (or selfsimilar solution) if there exist a positive
function a(t) and a 1parameter family of diffeomorphisms (p(t) : M — M
such that
(1.1) g(t) = a(tMt)*g(0).
Let dJltt denote the space of Riemannian metrics on a differentiable man
ifold M, and let 2)iff denote the group of diffeomorphisms of M. Consider
the quotient map IT : OKet — 9DTet/2Hff x R+, where E
+
acts by scalings.
One verifies that a Ricci soliton is a solution g (t) of the Ricci flow for which
n (g (£)) is independent of t, i.e., stationary.
We start by looking at what initial conditions give rise to Ricci solitons.
Differentiating (1.1) yields
(1.2)  2 Re (g(t)) = &(t)(p(t)*go + r(t)ip(t)* (Cxgo),
where go — ?(0), C denotes the Lie derivative, X is the timedependent
vector field such that X (p(i) (p)) = ^ (ip(t)(p)) for any p G M, and fr = 4*.
DEFINITION
1.2. For obvious reasons, we say that g(t) is expanding,
steady, or shrinking at a time £o if & (^o) is 0, — 0, or 0, respectively.
Since Rc(g(t)) = ^(£)*Rc(go)j w e c a n drop the pullbacks in (1.2) and
get
(1.3) 2Rc(#
0
) = cr{t)go + £x(t)#o,
where X (t) = a(t)X (t). Although go is independent of time, both &(t) and
X (t) may depend on time. For example, static Euclidean space
(Mn,
g{t)) =
(Kn,pCan)
is
a
stationary solution to the Ricci flow and, as such, is a steady
Ricci soliton; however, this solution may also be considered as a Ricci
soliton which expands or shrinks modulo diffeomorphisms. In particular,
given any function a (t) 0 with J(0) = 1, consider the diffeomorphisms
tp(t) :
Rn

Rn
defined by (p(t) (x) =
a{t)~l/2x
for x e
W1.
Then
^(^)*^can =
cr(0_1^can.
Since g(t) = ^can, we may rewrite this as
(1.4) 9(t)=gCa* = (r(tMt)*g{0).