Invariant conformal Killing forms on almost abelian Lie groups

• Herrera, A. C. ^[1] ; Origlia, M. ^[1]
  1. [1] Universidad Nacional de Córdoba

     Universidad Nacional de Córdoba

     Argentina

     
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 77, Fasc. 2, 2026, págs. 279-293
• Idioma: inglés
• DOI: 10.1007/s13348-025-00468-w
• Texto completo no disponible (Saber más ...)
• Resumen

  • We give a complete description of conformal Killing or conformal Killing–Yano (CKY) p-forms on almost abelian metric Lie algebras. In particular, we prove that if an n-dimensional almost abelian metric Lie algebra admits a non-parallel CKY p-form, then p=1 or p=n-1. In other words, any CKY p-form on a metric almost abelian Lie algebra is parallel for 2\le p\le n-2. We determine all metric almost abelian Lie algebras of dimension 4 (up to isometry) admitting non-parallel CKY p-forms. Moreover, we classify all Lie algebras with this property up to dimension 5, distinguishing also those cases where the associated simply connected Lie group admits lattices.

• Referencias bibliográficas
  • Alexandrova, I.A., Stepanov, S.E., Tsyganok, I.I.: Exterior differential forms on Riemannian symmetric spaces. Sci. Evol. 2(2), 49–53 (2017)
  • Andrada, A., Barberis, M.L., Dotti, I.: Invariant solutions to the conformal Killing–Yano equation on Lie groups. J. Geom. Phys. 94, 199–208...
  • Andrada, A., Barberis, M.L., Dotti, I., Ovando, G.: Product structures on four dimensional solvable Lie algebras. Homol. Homot. Appl. 7, 9–37...
  • Andrada, A., Dotti, I.: Conformal Killing–Yano 2-forms. Differ. Geom. Appl. 58, 103–119 (2018)
  • Andrada, A., Origlia, M.: Lattices on almost abelian Lie groups. Manuscr. Math. 155, 389–417 (2018)
  • Belgun, F., Moroianu, A., Semmelmann, U.: Killing forms on symmetric spaces. Differ. Geom. Appl. 24, 215–222 (2006)
  • Bock, C.: On low-dimensional solvmanifolds. Asian J. Math. 20, 199–262 (2016)
  • Del Barco, V., Moroianu, A.: Killing forms on 2-step nilmanifolds. J. Geom. Anal. 31, 863–887 (2021)
  • Figula, A., Nagy, P.: Isometry classes of simply connected nilmanifolds. J. Geom. Phys. 132, 370–381 (2018)
  • Freibert, M.: Cocalibrated structures on Lie algebras with a codimension one Abelian ideal. Ann. Glob. Anal. Geom. 42, 256 (2011). https://doi-org.sire.ub.edu/10.1007/s10455-012-9326-0
  • Herrera, A., Origlia, M.: A survey on invariant conformal Killing forms on Lie groups. Mat. Contemp. 60, 65-102.
  • Kashiwada, T.: On conformal Killing tensor. Natur. Sci. Rep. Ochanomizu Univ. 19, 67–74 (1968)
  • Lauret, J.: Homogeneous nilmanifolds of dimensions 3 and 4. Geom. Dedic. 68, 145–155 (1997)
  • Lee, J.B., Lee, K.B., Shin, J., Yi, S.: Unimodular groups of type R^{3}\rtimes R. J. Korean Math. Soc. 44, 1121–1137 (2007)
  • Malcev, A.: On solvable Lie algebras. Bull. Acad. Sci. URSS. Sr. Math. [Izvestia Akad. Nauk SSSR] 9, 329–356 (1945)
  • Medina, A., Revoy, P.: Lattices in symplectic Lie groups. J. Lie Theory 17, 27–39 (2007)
  • Milnor, J.: Curvature of left-invariant metrics on Lie groups. Adv. Math. 21, 293–329 (1976)
  • Moroianu, A., Semmelmann, U.: Killing forms on quaternion-Kähler manifolds. Ann. Global Anal. Geom. 28, 319–335 (2005); erratum 34, 431–432...
  • Moroianu, A., Semmelmann, U.: Twistor forms on Riemannian products. J. Geom. Phys. 58, 1343–1345 (2008)
  • Santillán, O.: Hidden symmetries and supergravity solutions. J. Math. Phys. 53, 043509 (2012)
  • Semmelmann, U.: Conformal Killing forms on Riemannian manifolds. Math. Z. 245, 503–527 (2003)
  • Semmelmann, U.: Killing forms on G_2 and Spin_7-manifolds. J. Geom. Phys. 56(9), 1752–1766 (2006)
  • Stepanov, S.E.: The vector space of conformal Killing forms on a Riemannian manifold. J. Math. Sci. 110, 2892–2906 (2002)
  • Stepanov, S.E., Jukl, M., Mikes, J.: On dimensions of vector spaces of conformal Killing forms. Algebra Geom. Math. Phys. 85, 495–507 (2014)
  • Sukilović, T.: Classification of left-invariant metrics on 4-dimensional solvable Lie groups. Theoret. Appl. Mech. 47, 181–204 (2020)
  • Tachibana, S.: On Killing tensors in a Riemannian space. Tohoku Math. J. 20, 257–264 (1969)
  • Yano, K.: Some remarks on tensor fields and curvature. Ann. Math. 55, 328–347 (1952)