The Bochner type curvature tensor of contact Riemannian structure (Q919632)

Let (M,\(\eta\),g) be a contact Riemannian manifold, with a contact form \(\eta\) and Riemannian metric g. One of the important problems in the study of contact manifolds is to find differential geometric properties which are independent of the choice of contact forms \(f\eta\), f being a positive function on M. The author studies gauge transformations of a contact Riemannian structure. On a contact Riemannian manifold, the author defines a Bochner type curvature tensor B for the subspace \(P_ x\) of the tangent space \(T_ xM\) to M at each point x, where \(P_ x\) is defined by \(\eta =0\). He proves the following theorem: The Bochner type curvature tensor B of a contact Riemannian manifold (M,\(\eta\),g) is invariant by gauge transformations (\(\eta\to {\tilde \eta}=f\eta)\) of contact Riemannian structures, if and only if the CR-structure corresponding to (\(\eta\),\(\phi\)) is integrable.