Ricci soliton homogeneous nilmanifolds

JORGE LAURET

MATHEMATISCHE ANNALEN

SPRINGER

Lugar: Berlin; Año: 2001 vol. 319 p. 715 - 733

0025-5831

We study a notion weakening the Einstein condition on a leftinvariant Riemannian metric $g$ on a nilpotent Lie group $N$.  Weconsider those metrics satisfying $\Ric_g=cI+D$ for some $c\in\RR$and some derivation $D$ of the Lie algebra $\ngo$ of $N$, where$\Ric_g$ denotes the Ricci operator of $\nil$.  This condition isequivalent to the metric $g$ to be a Ricci soliton.  We prove that aRicci soliton left invariant metric on $N$ is unique up to isometryand scaling.  The following characterization is also given: $\nil$is a Ricci soliton if and only if $\nil$ admits a metric standardsolvable extension whose corresponding standard solvmanifold$(S,\tilde{g})$ is Einstein.  This gives several families of newexamples of Ricci solitons.  By a variational approach, wefurthermore show that the Ricci soliton homogeneous nilmanifolds$\nil$ are precisely the critical points of a natural functionaldefined on a vector space which contains all the homogeneousnilmanifolds of a given dimension as a real algebraic set.