Annals of Global Analysis and Geometry

, Volume 51, Issue 3, pp 305–325 | Cite as

Curvature properties of metric nilpotent Lie algebras which are independent of metric

  • Grant Cairns
  • Ana Hinić Galić
  • Yuri NikolayevskyEmail author


This paper consists of two parts. First, motivated by classic results, we determine the subsets of a given nilpotent Lie algebra \(\mathfrak {g}\) (respectively, of the Grassmannian of two-planes of \(\mathfrak {g}\)) whose sign of Ricci (respectively, sectional) curvature remains unchanged for an arbitrary choice of a positive definite inner product on \(\mathfrak {g}\). In the second part we study the subsets of \(\mathfrak {g}\) which are, for some inner product, the eigenvectors of the Ricci operator with the maximal and with the minimal eigenvalue, respectively. We show that the closure of these subsets is the whole algebra \(\mathfrak {g}\), apart from two exceptional cases: when \(\mathfrak {g}\) is two-step nilpotent and when \(\mathfrak {g}\) contains a codimension one abelian ideal.


Metric nilpotent Lie algebra Ricci curvature Sectional curvature 

Mathematics Subject Classification

53C30 17B30 



The authors gratefully acknowledge the contribution of Marcel Nicolau (Barcelona).


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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsLa Trobe UniversityMelbourneAustralia

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