Solvable extensions of negative Ricci curvature of filiform Lie groups (Q2792257)

An area of continuing research in differential geometry is the determination of conditions, either necessary or sufficient, for a Lie group, or more generally, a homogeneous manifold, to admit a left-invariant metric of strictly negative Ricci curvature. The author obtains a necessary and sufficient condition for the existence of such a metric for any solvable Lie group with Lie algebra having filiform nilradical.NEWLINENEWLINEThis builds on previous work of the author and \textit{Yu. G. Nikonorov} [Math Z. 280, No. 1--2, 1--16 (2015; Zbl 1333.53071)] in which similar criteria were obtained for solvable Lie groups for which the nilradical of the Lie algebra was respectively abelian, Heisenberg or standard filiform. As the author explains in the introduction, it is known that no unimodular solvable Lie group admits a left-invariant metric of strictly negative Ricci curvature, as any unimodular Lie group with such a metric is necessarily semisimple and noncompact [\textit{I. Dotti Miatello} et al., Geom. Dedicata 17, No. 2, 207--218 (1984; Zbl 0549.53049)]. Hence, the conditions in the paper characterise a subset of nonunimodular solvable Lie groups. This necessarily consists of Lie groups which are not nilpotent.NEWLINENEWLINEIt is moreover observed at the conclusion of the article that by a result of \textit{Jorge Lauret} [in: Differential geometry and its applications. Proceedings of the 9th international conference on differential geometry and its applications, DGA 2004, Prague, Czech Republic, August 30--September 3, 2004. Prague: matfyzpress. 79--97 (2005; Zbl 1113.53033)], the Ricci tensor on an Einstein nilradical may be expressed in terms of the moment map of symplectic geometry. This suggests in itself an alternative approach to Theorem 1.3, the main theorem of the article under review, insofar as central to the present argument is the analysis of the Ricci operator on the metric nilradical. To this possible alternative approach still more promise is lent by the further observation that a key lemma, Lemma 3.4, is a special case of a more general result of the author [Geom. Dedicata 135, 87--102 (2008; Zbl 1145.53040), Lemma 2] which is itself derived from a convexity theorem for the image of the moment map [\textit{P. Heinzner} and \textit{H. Stötzel}, Math. Ann. 338, No. 1, 1--9 (2007; Zbl 1129.32015), Proposition 3].