Adapted complex polarizations on solvable Lie groups of dimension 3
Abstract
In this paper we investigate adapted complex polarizations associated with left-invariant Koszul connections on real Lie groups. Adapted complex polarizations arise as a natural generalization of a complex structure on a manifold. We investigate the existence of such polarizations on the tangent bundle of certain Lie groups with left-invariant connections. We use a practical criterion for the existence of global adapted complex polarizations in terms of the iℝ-completeness of the geodesic flow of the connection. In the Lie case, the Euler-Arnold vector field determines the behaviour of the geodesic flow. By combining what we know about the Euler-Arnold vector field of certain Lie groups and the iℝ-completeness criterion, we provide several examples of left-invariant ac-polarizations defined on the tangent bundle of Lie groups. Along with pseudo-Riemannian metric connections, we discuss biinvariant connections in several cases. Our focus is on solvable Lie groups, because non-solvable Lie groups are better understood. The main result of this paper is that we demonstrate several not locally symmetric but 𝒫-complete connections.
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BibTeX
@misc{forman2026adapted,
title = {{Adapted complex polarizations on solvable Lie groups of dimension 3}},
author = {Balázs Forman},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10937},
eprint = {2610.10937},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10937},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Adapted complex polarizations on solvable Lie groups of dimension 3
AU - Balázs Forman
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10937
UR - https://arxiv.org/abs/2610.10937
AB - In this paper we investigate adapted complex polarizations associated with left-invariant Koszul connections on real Lie groups. Adapted complex polarizations arise as a natural generalization of a complex structure on a manifold. We investigate the existence of such polarizations on the tangent bundle of certain Lie groups with left-invariant connections. We use a practical criterion for the existence of global adapted complex polarizations in terms of the iℝ-completeness of the geodesic flow of the connection. In the Lie case, the Euler-Arnold vector field determines the behaviour of the geodesic flow. By combining what we know about the Euler-Arnold vector field of certain Lie groups and the iℝ-completeness criterion, we provide several examples of left-invariant ac-polarizations defined on the tangent bundle of Lie groups. Along with pseudo-Riemannian metric connections, we discuss biinvariant connections in several cases. Our focus is on solvable Lie groups, because non-solvable Lie groups are better understood. The main result of this paper is that we demonstrate several not locally symmetric but 𝒫-complete connections.
ER -