Sard property for rank 2 polarizations in metabelian Lie groups

Enrico Le Donne; Antonio Lerario; Luca Nalon; Nicola Paddeu; Luca Rizzi

Sard property for rank 2 polarizations in metabelian Lie groups

Differential Geometry 2026-07-14 v1 Metric Geometry Optimization and Control

Authors: Enrico Le Donne , Antonio Lerario , Luca Nalon , Nicola Paddeu , Luca Rizzi

Abstract

We provide bounds on the dimension of the abnormal set for rank 2 polarizations on metabelian Lie groups, establishing the Sard property for the end-point map of such groups. We also obtain bounds for the dimension of the Goh-abnormal set for metabelian Lie groups where the codimension of the derived subgroup is at most 2, with no assumption on the rank of the polarization. We thus infer that these polarized groups, equipped with sub-Riemannian structures, satisfy the minimizing Sard property.

Keywords

Sard property abnormal set rank 2 polarization metabelian Lie group sub-Riemannian structure end-point map

Cite

@article{arxiv.2607.12530,
  title  = {Sard property for rank 2 polarizations in metabelian Lie groups},
  author = {Enrico Le Donne and Antonio Lerario and Luca Nalon and Nicola Paddeu and Luca Rizzi},
  journal= {arXiv preprint arXiv:2607.12530},
  year   = {2026}
}

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20 pages