Distributions and the Lie algebras their bases can generate

Abstract:

The problem is to determine when a smooth, $k$-dimensional distribution ${D^k}$ defined on an $n$-manifold ${M^n}$, locally admits a vector field basis which generates a nilpotent, solvable or even finite-dimensional Lie algebra. We show that for every $2 \leq k \leq n - 1$ there exists a (nonregular at $p \in {M^n}$) distribution ${D^k}$ on ${M^n}$ which does not locally (near $p$) admit a vector field basis generating a solvable Lie algebra. From classical results on the equivalence problem, it is shown that for $1 \leq k \leq 4$ and ${D^k}$ regular at $p \in {M^4}$, ${D^k}$ admits a local vector field basis generating a nilpotent Lie algebra.