On the pseudohermitian sectional curvature of a strictly pseudoconvex CR manifold

Abstract

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