- GENERALIZED FIXED POINTS 23
EXAMPLE 2.3. Let (JRn, gcan) denote Euclidean space with its standard
metric. Since the metric is flat, the trivial solution of (2.3) obtained by
choosing X = 0 lets us regard this as a steady Ricci soliton. But it may also
be regarded as an expanding gradient Ricci soliton, called the Gaussian
soliton, by taking .A = 1 and choosing the potential function
1 2
f (x) = 2 lxl ·In this section, we will prove the following observation.LEMMA 2.4. If (Mn,g(t)) is a solution of the Ricci flow {2.1) having
the special form {2.2), then there exists a vector field X on Mn such that
(Mn,go,X) solves {2.3). Conversely, given any solution (Mn,go,X) of
{2.3), there exist I -parameter families of scalars er (t) and diffeomorphisms
'l/Jt of Mn such that (Mn,g (t)) becom es a solution of the Ricci flow {2. 1)
when g (t) is defined by {2.2).
In words, this result says that there is a bijection between the families
of self-similar solutions and Ricci solitons which allows us to regard the
concepts as equivalent.
PROOF OF LEMMA 2.4. First suppose that (Mn,g(t)) is a solution of
the Ricci flow having the form (2.2). We may assume without loss of gener-
ality that er (0) = 1 and '1/Jo =id. Then we have-2 Re (go) = ~ g (t) I =er' (0) go+ Ly(o)go,
ut t=Owhere Y (t) is the family of vector fields generating the diffeomorphisms 'l/Jt.
This implies that go satisfies (2.3) with .A= ~er' (0) and X = Y (0).
Conversely, suppose that go satisfies (2.3). Define
er(t) ~ 1 +2.At,
and define a 1-parameter family of vector fields yt on Mn by
1
yt (x) ~ er (t) X (x).Let '!/Jt denote the diffeomorphisms generated by the family yt, where '!/Jo =
idMn, and define a smooth 1-parameter family of metrics on Mn by
g ( t) ~ er ( t) · 'lf;; (go).
Then g (t) has the special form (2.2). The computation:tg = ~~ · 'lf;; (go) +er (t) · 'lf;; (£ytgo) = 'lf;; (2.Ago + L:xgo).
implies by (2.3) that
a
otg = 'lf;; (- 2 Re [go]) = -2 Re [g] ,hence that g (t) is a solution of the Ricci flow.^0