- WARPED PRODUCTS AND 2-DIMENSIONAL SOLITONS 15
Likewise, taking b = 0, we have the ODE w' = aw^2. If a = -a^2 , then
w (r) = °' 2 ;+c, f (r) = -2 log ( a^2 r + C) and K = (r+c!- 2 ) 2 •EXERCISE 1.22. Show that if w is a solution to (1.42), i.e., w' -aw^2 = b,
then v ~ 1/w satisfies
v' + bv^2 =-a.
Related to the above exercise, in Section 9 of this chapter we shall see
Buscher duality exhibited in the above solitons.EXERCISE 1.23. Determine all of the solutions of the rotationally sym-
metric steady gradient Ricci soliton equation on a surface.3.4. A metric transformation on surfaces related to circle ac-
tions. It is interesting to search for duality transformations, besides Buscher
duality, for Ricci solitons. One transformation for rotationally symmetric
metrics on surfaces is the following, which is related to the study of col-
lapsing sequences of solutions of the Ricci flow on 3-manifolds. Given a
rotationally symmetric metric g = dr^2 +w (r)^2 de^2 on a surface M^2 , we may
define another metric on M^2 by
g ~ dr2 + w (r)2 de2.
a^2 w (r)^2 + 1A more general transformation is given by Cheeger [69]. The inverse trans-
formation is
w (r)^2
9 = dr^2 + de^2
· l-a^2 w(r)^2 '
which is defined as long as 0 < w (r) < 1/a.
EXERCISE 1.24. Consider the 51 action on (M^2 ,g) x 51 (1/a), where
51 (p) ~ IR/27rp'!L, defined by
¢(r,e,'J7)~(r,e+¢,'17+¢), ¢E5^1.
Show that g is the quotient metric on [(M^2 ,g) x 51 (1/a)] /5^1.
SOLUTION. See Proposition 2 of [103].
We now consider some examples. The inverse transformation of the
cigar,.__, g cig = dr (^2) + 1. 2( ) 2
(^2) a smh ar de ,
is the hyperbolic metric of constant curvature K = -a^2. The transformation
of the exploding soliton,
gxpd ~ = d r^2 + 2 1. sm 2 ( ar )de2 ,
a
is the spherical metric of constant curvature K = a^2. Since gxpd is only
defined for 0 < r < 7r/ (2a), we see that §xpd is a hemisphere. In summary,
we have the following types of examples: