Rigid Two-Step Nilpotent Lie Groups Relative to Multicontact Structures

Abstract

Let $\{X_i,Y_k\}$ denote a fixed basis of the Lie algebra $\mathfrak{n}$ of a connected and simply connected nilpotent Lie group $N$ of step two. Under a technical assumption on $\{X_i,Y_k\}$, we prove that the Lie algebra $\mathcal{M}$ of vector fields $V$ on $N$ that satisfy $[V,X_i]={\lambda }_i{X}_i$ is finite dimensional, a property that we refer to as rigidity. Our proof allows the explicit count of $\dim \mathcal{M}$.