The Lie algebra generated by gradient vector fields

Bettina Kazandjian; Omar Mohsen; Eugenio Pozzoli

differential geometry

The Lie algebra generated by gradient vector fields

arXiv:2607.26890

summary

The paper shows that on any smooth compact manifold with a non-degenerate symmetric bilinear form, every smooth vector field can be expressed as a finite linear combination of iterated Lie brackets of gradient vector fields.

Abstract

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

Topics & keywords

#lie algebras #gradient vector fields #smooth manifolds #vector fields #bilinear formsLie bracketgradient vector fieldcompact manifoldnon-degenerate symmetric bilinear formsmooth vector field