An integral inequality for normal contact Riemannian manifolds and its applications (Q1819748)

A compact Sasakian manifold \((M,g)\) is said to be cohomologically Einsteinian if the 2-form \(\tilde S(X,Y)=S(X,\phi Y)\) is exact, where S and \(\phi\) denote the Ricci curvature tensor and the almost contact structure tensor of \((M,g)\), respectively. It is proven here that the upper bound of the volume of a compact Sasakian manifold can be expressed by an integral formula involving S and the scalar curvature \(\rho\) only. Utilizing this formula the author derives the following theorem: If a compact cohomologically Einsteinian Sasakian manifold \((M,g)\) of \(2n+1\) dimensions satisfies the condition \(\rho =2n(2n+1)\), then (M,g) itself is an Einstein space.