On the pseudohermitian sectional curvature of a strictly pseudoconvex CR manifold

We show that the pseudohermitian sectional curvature $H_\theta (\sigma)$ of a contact form $\theta$ on a strictly pseudoconvex CR manifold $M$ measures the difference between the lengths of a circle in a plane tangent at a point of $M$ and its projection on $M$ by the exponential map associated to the Tanaka-Webster connection of $(M , \theta)$. Any Ssakian manifold $(M, \theta)$ whose pseudohermitian sectional curvature $K_\theta(\sigma)$ is a point function is shown to be Tanaka-Webster flat, and hence a Sasakian space form of $\varphi$-sectional curvature $c=-3$. We show that the Lie algebra $\iota (M, \theta)$ of all infinitesimal pseudohermitian transformations on a strictly pseudoconvex CR manifold $M$ of CR dimension $n$ has dimension $\leq (n+1)^2$ and if $\mathrm{dim}_{\mathbb R} \iota (M , \theta)=(n+1)^2$ then $H_\theta (\sigma) = constant.