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Locally conformally product structures on solvmanifolds

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Abstract

We study left invariant locally conformally product structures on simply connected Lie groups and give their complete description in the solvable unimodular case. Based on previous classification results, we then obtain the complete list of solvable unimodular Lie algebras up to dimension 5 which carry LCP structures, and study the existence of lattices in the corresponding simply connected Lie groups.

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Acknowledgements

A. M. and V. dB. thank the French-Brazilian network in Mathematics and the CAPES-COFECUB project 895/18 for financial support, and are grateful to the Mathematical Department of the Universitá degli Studi di Perugia, where this work was initiated. A. A. would like to thank the Laboratoire de Mathématiques d’Orsay for hospitality. A. A. is partially supported by CONICET and SECYT-UNC (Argentina). V. dB. is partially supported by FAPESP Grant 2021/09197-8. The authors are supported by MATHAMSUD Regional Program 21-MATH-06.

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Authors and Affiliations

  1. FAMAF, Universidad Nacional de CĂłrdoba and CIEM-CONICET, Av. Medina Allende s/n, Ciudad Universitaria, X5000HUA, CĂłrdoba, Argentina

    Adrián Andrada

  2. Instituto de Matemática, Estatística e Computação Científica, Universidade Estadual de Campinas, Rua Sergio Buarque de Holanda, 651, Cidade Universitaria Zeferino Vaz, Campinas, São Paulo, 13083-859, Brazil

    Adrián Andrada & Viviana del Barco

  3. Université Paris-Saclay, CNRS, Laboratoire de mathématiques d’Orsay, 91405, Orsay, France

    Andrei Moroianu

  4. Institute of Mathematics “Simion Stoilow” Romanian Academy, 21 Calea Grivitei, 010702, Bucharest, Romania

    Andrei Moroianu

Authors
  1. Adrián Andrada
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  2. Viviana del Barco
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  3. Andrei Moroianu
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Correspondence to Adrián Andrada.

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Appendix A: Tables of low-dimensional LCP Lie algebras

Appendix A: Tables of low-dimensional LCP Lie algebras

This appendix contains the structure constants of the unimodular solvable LCP Lie algebras up to dimension 5. In each table, the 3rd column gives the possible dimensions of the flat factor of an LCP structure, as determined in Sect. 7. The 4th column says whether the corresponding simply connected Lie group has lattices or not. Notice that when the Lie algebra belongs to a family, it is in general not possible to determine explicitly the set of parameters for which the corresponding simply connected Lie groups admit lattices (except when one can show that none of them does).

Table 1 3-dimensional LCP Lie algebra
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Table 2 4-dimensional LCP Lie algebras
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Table 3 5-dimensional LCP Lie algebras
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Andrada, A., del Barco, V. & Moroianu, A. Locally conformally product structures on solvmanifolds. Annali di Matematica (2024). https://doi.org/10.1007/s10231-024-01449-9

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