On the pseudohermitian sectional curvature of a strictly pseudoconvex CR manifold

We show that the pseudohermitian sectional curvature Hθ(σ)Hθ(σ) of a contact form θ 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,θ)(M,θ). Any Sasakian manifold (M,θ)(M,θ) whose pseudohermitian sectional curvature Kθ(σ)Kθ(σ) is a point function is shown to be Tanaka–Webster flat, and hence a Sasakian space form of φ  -sectional curvature c=−3c=−3. We show that the Lie algebra i(M,θ)i(M,θ) of all infinitesimal pseudohermitian transformations on a strictly pseudoconvex CR manifold M of CR dimension n   has dimension ⩽(n+1)2⩽(n+1)2 and if dimRi(M,θ)=(n+1)2dimRi(M,θ)=(n+1)2 then Hθ(σ)=Hθ(σ)= constant.