A subspace of the algebra of infinitesimal CR automorphisms

Question

Let $(M^{2n+1}, D, J)$ be a strictly pseudoconvex CR sturcture on a compact $2n+1$-dimensional manifold, where $D$ is a nonintegrable distribution of codimension 1. The algebra of the infinitesimal CR transformation is then $$ \mathfrak{cr}(D, J) = \{X \in \Gamma(T M): [X, D] \subseteq D, L_X J = 0 \} $$ where the Lie derivative $L_X J$ makes sense since the bracket of $X$ with elements in $D$ still lies in $D$.

My question is: What is the space $\mathfrak{cr}(D, J) \cap \Gamma(D)$? Is it always zero?

thanks

David

Answer 1

An even stronger statement is true: If $[X,D]\subset D$ and $X$ belongs to $\Gamma(D)$, then $X = 0 $. The reason is that, because $D$ is a contact $2n$-plane field (since your CR structure is strictly pseudoconvex), if $X$ were nonzero, there would have to be another vector field $Y\in\Gamma(D)$ such that $[X,Y]$ does not lie in $D$ at any point.

Thus, a fortiori, ${\frak{cr}}(D,J)\cap\Gamma(D)=\{0\}$.