The Lie algebra generated by gradient vector fields

Bettina Kazandjian; Omar Mohsen; Eugenio Pozzoli

The Lie algebra generated by gradient vector fields

Differential Geometry 2026-07-29 v1 Optimization and Control

Authors: Bettina Kazandjian , Omar Mohsen , Eugenio Pozzoli

Abstract

Let $M$ be a smooth compact manifold, and $g$ a smooth non-degenerate symmetric bilinear form on $TM$ . We prove that every smooth vector field can be written as a finite linear combination of iterated Lie brackets of gradient vector fields.

Cite

@article{arxiv.2607.26890,
  title  = {The Lie algebra generated by gradient vector fields},
  author = {Bettina Kazandjian and Omar Mohsen and Eugenio Pozzoli},
  journal= {arXiv preprint arXiv:2607.26890},
  year   = {2026}
}