Rigidity of Nilpotent Lie Foliations: Cohomological Obstructions and Classification

TL;DR

This paper introduces a cohomological approach to study the rigidity of nilpotent Lie foliations, identifying obstructions to deformation and classifying foliations up to codimension six.

Contribution

It develops a new cohomological framework for analyzing rigidity, providing algebraic criteria and a classification for low codimension nilpotent Lie foliations.

Findings

Obstructions to deformation lie in specific cohomology groups.

A criterion for rigidity in generalized Heisenberg group foliations.

Complete classification of nilpotent Lie foliations up to codimension six.

Abstract

In this article, we develop a systematic cohomological framework for the study of the rigidity of nilpotent Lie foliations with respect to solvable deformations. We introduce the deformation complex associated to a pair of Lie algebras $(\mathfrak{g}, \mathfrak{h})$ and show that the main obstruction to deforming a nilpotent Lie foliation into a non-nilpotent solvable foliation lies in the cohomology group $H^2(\mathfrak{g},\mathfrak{g}/[\mathfrak{g},\mathfrak{g}])$. We establish a necessary and sufficient algebraic criterion for rigidity within the family of foliations modelled on the generalized Heisenberg groups $H_{2k+1}$. This result unifies and generalizes the construction of Dathe--Ndiaye (2012) as well as its subsequent extensions. We complete the article with a full classification of nilpotent Lie foliations of codimension at most six according to their deformation behaviour.