Parallel skew-symmetric tensors on 4-dimensional metric Lie algebras
DOI:
https://doi.org/10.33044/revuma.2451Abstract
We give a complete classification, up to isometric isomorphism and scaling, of 4-dimensional metric Lie algebras $(\mathfrak{g},\langle\cdot,\cdot\rangle)$ that admit a non-zero parallel skew-symmetric endomorphism. In particular, we distinguish those metric Lie algebras that admit such an endomorphism which is not a multiple of a complex structure, and for each of them we obtain the de Rham decomposition of the associated simply connected Lie group with the corresponding left invariant metric. On the other hand, we find that the associated simply connected Lie group is irreducible as a Riemannian manifold for those metric Lie algebras where each parallel skew-symmetric endomorphism is a multiple of a complex structure.
Downloads
References
A. Andrada, M. L. Barberis, I. G. Dotti, and G. P. Ovando, Product structures on four dimensional solvable Lie algebras, Homology Homotopy Appl. 7 no. 1 (2005), 9–37. MR Zbl Available at http://projecteuclid.org/euclid.hha/1139839504.
A. L. Besse, Einstein Manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 10, Springer-Verlag, Berlin, 1987. DOI MR Zbl
A. Fino, Almost Kähler 4-dimensional Lie groups with $J$-invariant Ricci tensor, Differential Geom. Appl. 23 no. 1 (2005), 26–37. DOI MR Zbl
A. C. Herrera, Estructuras Killing-Yano invariantes en variedades homogéneas, Ph.D. thesis, Universidad Nacional de Córdoba, Argentina, 2018. Available at http://hdl.handle.net/11086/6554.
G. R. Jensen, Homogeneous Einstein spaces of dimension four, J. Differential Geometry 3 (1969), 309–349. MR Zbl Available at http://projecteuclid.org/euclid.jdg/1214429056.
G. P. Ovando, Invariant pseudo-Kähler metrics in dimension four, J. Lie Theory 16 no. 2 (2006), 371–391. MR Zbl
P. Petersen, Riemannian Geometry, third ed., Graduate Texts in Mathematics 171, Springer, Cham, 2016. DOI MR Zbl
C. T. C. Wall, Geometries and geometric structures in real dimension $4$ and complex dimension $2$, in Geometry and Topology (College Park, Md., 1983/84), Lecture Notes in Math. 1167, Springer, Berlin, 1985, pp. 268–292. DOI MR Zbl
C. T. C. Wall, Geometric structures on compact complex analytic surfaces, Topology 25 no. 2 (1986), 119–153. DOI MR Zbl
Downloads
Published
Issue
Section
License
Copyright (c) 2023 Andrea C. Herrera
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal. The Journal may retract the paper after publication if clear evidence is found that the findings are unreliable as a result of misconduct or honest error.
