36 1. RICCI SOLITONS
6.2. Construction. The first explicit examples of Ricci soliton struc-
tures on Lie groups were constructed by Baird and Daniela [15] and in-
dependently by Lott [256]. We discuss [15] in this section· and [256] in
subsection 6.3, below.
Baird and Daniela discovered soliton structures on 3-dimensional man-
ifolds by studying semiconformal maps to Riemannian surfaces. Signifi-
cantly, their constructions give the first known examples of nongradient
soliton structures. We will only describe two of their conclusions, referring
the reader to [15] for details of the method.
EXAMPLE 1.54. Let N denote the 3-dimensional Heisenberg group ni1^3.
Recall that ·N may be represented as the group of upper-triangular ·ma.trices
under matrix multiplication.
It is a general fact^14 that any simply-connected nilpotent Lie group is
diffeomorphic to JR.n. So give JR.^3 its standard coordinates (x1, x2, x3) and
define the frame field
{) {) {) {)F1=28x1' F2=2(8x2 -x18x3), F3=28x3·
It is easy to check that all brackets [Fi, Fj] vanish. except [F1, F2] = -2F3,
hence that (F 1 , F 2 , F 3 ) is a nil^3 -geometry frame.^15 The connection 1-forms
may be displayed as(V' F 1 F1 V' F 1 F2
V' p 2 F1 V' p 2 F2
V' F 3 F1 V' p 3 F2
Using the dual field
1 1w =
2
dx1,1
w^2 = 2 dx2,
define a left-invariant metric on N by-F3
0-F1
g = 4 (w^1 @ w^1 + w^2 @ w^2 + w^3 @ w^3 ).
Recalling the standard formula(R(X, Y)Y, X) = ~ l(adX)Y +(ad Y)Xl2 - ((adX) X, (ad Y)Y)
3 2 1 1- 4 l[X, Y]I - 2 ([[X, Y], Y], X) -
2
([[Y, X],X], Y),it is straightforward to compute that
Rc(g) = -2(w^1 @ w^1 ) - 2(w^2 @ w^2 ) + 2(w^3 @ w^3 ).(^14) The exponential map of a connected, simply-connected, nilpotent Lie group is a
diffeomorphism. For instance, see [200].
(^15) See Volume One, Chapter 1, Sections 3-4.