- HOMOGENEOUS EXPANDING SOLITONS 37
Define a vector field
1 1 1
X = -2x1F1 - 2x2F2 - (2x1x2 + x3)F3.A calculation shows that the coordinates (\JiXJ) of \7 X = \JiXJ wi Q9 Fj
correspond to the matrix(-1 -(!x1x2 + x3)
(\JiXj) = !x1~2 + X3 -;:1
2X2 -2X1It is then easy to see that
-2Rc(g) = £'xg + 3g,
hence that (N, g, X) is a Ricci soliton structure.
REMARK 1.55. Because the 1-form metrically dual to X is not closed,it follows that X #grad f for any soliton potential function f.
REMARK 1.56. Compact locally homogeneous manifolds with ni1^3 geom-etry occur as mapping tori of YA : T^2 -----t T^2 induced by A = (~ ~) E
SL(2, Z) with k # 0. The left-invariant metric g is compatible with any
compact quotient, but the soliton structure is never compatible with com-
pactification. Indeed, the scalar curvature of g is R = -1/2, while every
compact Ricci soliton of nonpositive scalar curvature is Einstein.
EXAMPLE 1.57. Let S denote the simply-connected 3-dimensional solv-
able Lie group sol^3 =IR ><1 ffi.^2 , where the action of u E IR sends (v, w) E ffi.^2
to (euv, e-uw). One may also regard S as the group of rigid motions of
Minkowski 2-space.
S is diffeomorphic to ffi.^3. In standard coordinates ( x1, ;;2, X3) on ffi.^3 ,
define the frame field
F1=2
8(^8) , F2 = 2(e-x (^1) ~+ex 1
8
(^8) ), F3 = 2(e-xi~_:_ex (^1) ~).
X1 ax2 X3 ax2 ax3
Its bracket relations are [F1, F2] = -2F3, [F2, F3] = 0, and [F3, F1] = 2F2.
So (F 1 , F 2 , F 3 ) is a sol^3 -geometry frame. The connection 1-forms may be
displayed as
Using the dual field
1 1
w =
2
dx1,define a left-invariant metric on S by
0
0
-4F1g = 4(w^1 Q9 w^1 ) + 8(w^2 Q9 w^2 ) + 8(w^3 Q9 w^3 ).