34 1. RICCI SOLITONS
One begins with the observation that conditions (1.64)-(1.66) are equiv-
alent for a nilpotent group.
LEMMA 1.47. Let N be a simply-connected nilpotent Lie group with Lie
algebra n and a left-invariant metric g. Then there exist .\ E IR and X E nsolving the Ricci soliton structure equation (1.64) if and only if there exist
.\ E IR and a derivation D E ker( 81) solving (1.66).
DEFINITION 1.48. A standard metric solvable extension of ( n, g) is a
pair (s, g), where .s = aEBn is a solvable Lie algebra such that [·, ·lslnxn = [·, ·]n
and g is an inner product of standard type such that [.s,.s]s = n =a..l and
9[nxn = g.
Recalling Definition 1.45, note that for g to be of standard type is equiv-
alent to a being abelian. The following result relates Ricci soliton structures
on nilpotent Lie groups with Einstein metrics of standard type on solvable
Lie groups.
THEOREM 1.49. Let N be a simply-connected nilpotent Lie group with
Lie algebra n and a left-invariant metric g. Then ( N, g) admits a Riccisoliton structure if and only if ( n, g) admits a standard metric solvable ex-
tension (.s = aEBn, g) such that the simply-connected solvable Lie group (S, g)
is Einstein.Although its proof is nonconstructive, Theorem 1.49 is a very useful
criterion. For example, any generalized Heisenberg group [25] and many
other two-step nilpotent Lie groups admit a Ricci soliton structure. On
the other hand, if n is characteristically nilpotent, then N admits no such
structure. (See [244] for more examples.)
If a Ricci soliton structure does exist on N, it is essentially unique.
THEOREM 1.50. Let N be a simply-connected nilpotent Lie group with
Lie algebra n. If g and g' are both Ricci soliton metrics, then there exist
a> 0 and rt E Aut(n) such that g^1 = art(g).
REMARK 1.51. If one regards a Ricci soliton structure as a 3-tuple
(N, g, X) satisfying (1.64), then uniqueness of X must be understood mod-
ulo addition of a Killing vector field. (See Example 1.57 below.)
Some partial answers are known regarding the existence of Ricci soliton
structures on broader classes of Lie groups. For example, if G is semisim-
ple, then condition (1.66) implies condition (1.64). However, (1.66) has no
solutions other than Einstein metrics:
THEOREM 1.52. A left-invariant metric g on a simply-connected semisim-
ple Lie group G satisfies (1.66) if and only if D = 0, hence if and only if g
is Einstein.
It is not known in general whether or not a noncompact, semisimple Lie
group admits an Einstein metric.