Conformal Killing forms on 2-step nilpotent Riemannian Lie groups
-
Viviana del Barco
and
Andrei Moroianu
Abstract
We study left-invariant conformal Killing 2- or 3-forms on simply connected 2-step nilpotent Riemannian Lie groups. We show that if the center of the group is of dimension greater than or equal to 4, then every such form is automatically coclosed (i.e. it is a Killing form). In addition, we prove that the only Riemannian 2-step nilpotent Lie groups with center of dimension at most 3 and admitting left-invariant non-coclosed conformal Killing 2- and 3-forms are the following: The Heisenberg Lie groups and their trivial 1-dimensional extensions, endowed with any left-invariant metric, and the simply connected Lie group corresponding to the free 2-step nilpotent Lie algebra on 3 generators, with a particular 1-parameter family of metrics. The explicit description of the space of conformal Killing 2- and 3-forms is provided in each case.
Acknowledgements
We would like to thank the anonymous referee for a very careful reading of the first version of the manuscript, which enabled us to do several corrections and to simplify some arguments.
References
[1] A. Andrada and I. G. Dotti, Conformal Killing–Yano 2-forms, Differential Geom. Appl. 58 (2018), 103–119. 10.1016/j.difgeo.2018.01.003Search in Google Scholar
[2] A. Andrada and I. G. Dotti, Killing–Yano 2-forms on 2-step nilpotent Lie groups, Geom. Dedicata 212 (2021), 415–424. 10.1007/s10711-020-00564-0Search in Google Scholar
[3] V. Apostolov, D. M. J. Calderbank and P. Gauduchon, Hamiltonian 2-forms in Kähler geometry. I. General theory, J. Differential Geom. 73 (2006), no. 3, 359–412. 10.4310/jdg/1146169934Search in Google Scholar
[4] V. Apostolov, D. M. J. Calderbank, P. Gauduchon and C. W. Tønnesen Friedman, Hamiltonian 2-forms in Kähler geometry. II. Global classification, J. Differential Geom. 68 (2004), no. 2, 277–345. 10.4310/jdg/1115669513Search in Google Scholar
[5] V. Apostolov, D. M. J. Calderbank, P. Gauduchon and C. W. Tønnesen Friedman, Hamiltonian 2-forms in Kähler geometry. III. Extremal metrics and stability, Invent. Math. 173 (2008), no. 3, 547–601. 10.1007/s00222-008-0126-xSearch in Google Scholar
[6] M. L. Barberis, I. G. Dotti and O. Santillán, The Killing–Yano equation on Lie groups, Classical Quantum Gravity 29 (2012), no. 6, Article ID 065004. 10.1088/0264-9381/29/6/065004Search in Google Scholar
[7] F. Belgun, A. Moroianu and U. Semmelmann, Killing forms on symmetric spaces, Differential Geom. Appl. 24 (2006), no. 3, 215–222. 10.1016/j.difgeo.2005.09.007Search in Google Scholar
[8] V. del Barco and A. Moroianu, Higher degree Killing forms on 2-step nilmanifolds, J. Algebra 563 (2020), 251–273. 10.1016/j.jalgebra.2020.07.020Search in Google Scholar
[9] V. del Barco and A. Moroianu, Killing forms on 2-step nilmanifolds, J. Geom. Anal. 31 (2021), no. 1, 863–887. 10.1007/s12220-019-00304-1Search in Google Scholar
[10] P. Eberlein, Geometry of 2-step nilpotent groups with a left invariant metric, Ann. Sci. Éc. Norm. Supér. (4) 27 (1994), no. 5, 611–660. 10.24033/asens.1702Search in Google Scholar
[11] A. C. Herrera and M. Origlia, Invariant conformal Killing–Yano 2-forms on five dimensional Lie groups, preprint (2020), https://arxiv.org/abs/2012.11054. 10.1007/s12220-022-00951-xSearch in Google Scholar
[12] T. Kashiwada, On conformal Killing tensor, Natur. Sci. Rep. Ochanomizu Univ. 19 (1968), 67–74. Search in Google Scholar
[13] A. Moroianu, Killing vector fields with twistor derivative, J. Differential Geom. 77 (2007), no. 1, 149–167. 10.4310/jdg/1185550818Search in Google Scholar
[14] A. Moroianu and U. Semmelmann, Twistor forms on Kähler manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2 (2003), no. 4, 823–845. Search in Google Scholar
[15] A. Moroianu and U. Semmelmann, Killing forms on quaternion-Kähler manifolds, Ann. Global Anal. Geom. 28 (2005), no. 4, 319–335. 10.1007/s10455-005-1147-ySearch in Google Scholar
[16] A. Moroianu and U. Semmelmann, Twistor forms on Riemannian products, J. Geom. Phys. 58 (2008), no. 10, 1343–1345. 10.1016/j.geomphys.2008.05.007Search in Google Scholar
[17] P.-A. Nagy and U. Semmelmann, Conformal Killing forms in Kähler geometry, preprint (2020), https://arxiv.org/abs/2003.09184. 10.1215/00192082-10088173Search in Google Scholar
[18] A. M. Naveira and U. Semmelmann, Conformal Killing forms on nearly Kähler manifolds, Differential Geom. Appl. 70 (2020), Article ID 101628. 10.1016/j.difgeo.2020.101628Search in Google Scholar
[19] U. Semmelmann, Conformal Killing forms on Riemannian manifolds, Math. Z. 245 (2003), no. 3, 503–527. 10.1007/s00209-003-0549-4Search in Google Scholar
[20]
U. Semmelmann,
Killing forms on
[21] S.-I. Tachibana, On conformal Killing tensor in a Riemannian space, Tohoku Math. J. (2) 21 (1969), 56–64. 10.2748/tmj/1178243034Search in Google Scholar
[22] S. Yamaguchi, On a Killing p-form in a compact Kählerian manifold, Tensor (N. S.) 29 (1975), no. 3, 274–276. Search in Google Scholar
[23] K. Yano, Some remarks on tensor fields and curvature, Ann. of Math. (2) 55 (1952), 328–347. 10.2307/1969782Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Indecomposable involutive solutions of the Yang–Baxter equation of multipermutational level 2 with abelian permutation group
- Extrapolation to product Herz spaces and some applications
- The exact number of orthogonal exponentials on the spatial Sierpinski gasket
- A note on special cubic fourfolds of small discriminants
- Strong multiplicity one for Siegel cusp forms of degree two
- Gross–Prasad periods for reducible representations
- LS-category of moment-angle manifolds and higher order Massey products
- Estimates of heat kernels of non-symmetric Lévy processes
- Global continuity of variational solutions weakening the one-sided bounded slope condition
- Finitary birepresentations of finitary bicategories
- Solvability for nonlocal boundary value problems with generalized 𝑝-Laplacian on an unbounded domain
- Conformal Killing forms on 2-step nilpotent Riemannian Lie groups
- Simply transitive NIL-affine actions of solvable Lie groups
- Mahler measure of 3D Landau–Ginzburg potentials
