Critical associated metrics on contact manifolds. II

Abstract

Abstract The study of the integral of the scalar curvature, ∫ M RdV g , as a function on the set of all Riemannian metrics of the same total volume on a compact manifold is now classical, and the critical points are the Einstein metrics. On a compact contact manifold we consider this and ∫ M ( R − R * − 4 n 2 ) dv , with R * the *-scalar curvature, as functions on the set of metrics associated to the contact structure. For these integrals the critical point conditions then become certain commutativity conditions on the Ricci operator and the fundamental collineation of the contact metric structure. In particular, Sasakian metrics, when they exist, are maxima for the second function.