Killing Forms on 2-Step Nilmanifolds

  • Viviana del BarcoEmail author
  • Andrei Moroianu


We study left-invariant Killing k-forms on simply connected 2-step nilpotent Lie groups endowed with a left-invariant Riemannian metric. For \(k=2,3\), we show that every left-invariant Killing k-form is a sum of Killing forms on the factors of the de Rham decomposition. Moreover, on each irreducible factor, non-zero Killing 2-forms define (after some modification) a bi-invariant orthogonal complex structure and non-zero Killing 3-forms arise only if the Riemannian Lie group is naturally reductive when viewed as a homogeneous space under the action of its isometry group. In both cases, \(k=2\) or \(k=3\), we show that the space of left-invariant Killing k-forms of an irreducible Riemannian 2-step nilpotent Lie group is at most one-dimensional.


Killing forms 2-step nilpotent Lie groups Naturally reductive homogeneous spaces 

Mathematics Subject Classification

53D25 22E25 53C30 



We would like to thank the anonymous referee for several pertinent remarks and suggestions, which led to the low-dimensional classification results in Theorems 4.14 and 5.14, and to the global improvement of the presentation.


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© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques d’OrsayUniversidad Nacional de Rosario, CONICETRosarioArgentina
  2. 2.Univ. Paris-SudOrsayFrance
  3. 3.Laboratoire de Mathématiques d’OrsayUniv. Paris-Sud, CNRS, Université Paris-SaclayOrsayFrance

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