On the pseudohermitian sectional curvature of a strictly pseudoconvex CR manifold

• Published in: Differential geometry and its applications 2007-12, Vol.25 (6), p.612-631
• Main Author: Barletta, Elisabetta
• Format: Article
• Language: English
• Subjects: CR manifold; Jacobi field; Pseudohermitian sectional curvature; Tanaka–Webster connection
• ISSN: 0926-2245; 1872-6984
• DOI: 10.1016/j.difgeo.2007.06.009

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.