Rigidity and Affine Deformations of Foliations on Nilpotent and Solvable Contact Lie Algebras
Abstract
Let g be a Lie algebra of dimension 2n+1, let beta be a contact form, and let alpha be a nonzero integrable 1-form. We study when the affine path alpha_t = (1-t)alpha + t beta consists of contact forms with the orientation determined by beta \wedge (d beta)^n. We first establish an algebraic criterion: affine deformability is equivalent to the non-negativity of a scalar obtained from the interaction form Theta(alpha, beta) = alpha \wedge (d beta)^n + n beta \wedge d alpha \wedge (d beta)^{n-1}. For unimodular Lie algebras this scalar is (n+1)alpha(Z_beta), where Z_beta is the Reeb vector. We also derive a formula involving the modular character tau(X) = tr(ad_X) in the non-unimodular case. On nilpotent contact Lie algebras, every integrable 1-form is closed; using the standard structural properties of nilpotent contact Lie algebras, this implies that every left-invariant foliation is an affine origin for every fixed contact form. An explicit three-dimensional example illustrates that the modular term can yield affine deformability even when alpha(Z_beta) = 0. We discuss the implications for left-invariant forms on Lie groups and compact quotients. We do not claim a general obstruction example in the non-unimodular case; establishing such an example requires a separate verified construction.
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BibTeX
@misc{khoule2026rigidity,
title = {{Rigidity and Affine Deformations of Foliations on Nilpotent and Solvable Contact Lie Algebras}},
author = {Cheikh Khoule and Rafik Khalfi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35940},
eprint = {2610.35940},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35940},
}
RIS
TY - UNPB
TI - Rigidity and Affine Deformations of Foliations on Nilpotent and Solvable Contact Lie Algebras
AU - Cheikh Khoule
AU - Rafik Khalfi
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.35940
UR - https://arxiv.org/abs/2610.35940
AB - Let g be a Lie algebra of dimension 2n+1, let beta be a contact form, and let alpha be a nonzero integrable 1-form. We study when the affine path alpha_t = (1-t)alpha + t beta consists of contact forms with the orientation determined by beta \wedge (d beta)^n. We first establish an algebraic criterion: affine deformability is equivalent to the non-negativity of a scalar obtained from the interaction form Theta(alpha, beta) = alpha \wedge (d beta)^n + n beta \wedge d alpha \wedge (d beta)^{n-1}. For unimodular Lie algebras this scalar is (n+1)alpha(Z_beta), where Z_beta is the Reeb vector. We also derive a formula involving the modular character tau(X) = tr(ad_X) in the non-unimodular case. On nilpotent contact Lie algebras, every integrable 1-form is closed; using the standard structural properties of nilpotent contact Lie algebras, this implies that every left-invariant foliation is an affine origin for every fixed contact form. An explicit three-dimensional example illustrates that the modular term can yield affine deformability even when alpha(Z_beta) = 0. We discuss the implications for left-invariant forms on Lie groups and compact quotients. We do not claim a general obstruction example in the non-unimodular case; establishing such an example requires a separate verified construction.
ER -