Second Variation of the 'Total Scalar Curvature' on Contact Manifolds

摘要

Let M2m] be a compact contact manifold and A the set of associated metrics. Using the scalar curvature R and the *-scalar curvature R*, in [5] we defined the "total scalar curvature", by I(g) = fM j(R + R* + 4n(n + ))dV and showed that the critical points of I(g) on A are the K-contact metrics, i.e. metrics for which the characteristic vector field is Killing. In this paper we compute the second variation of I(g) and prove that the index of I(g) and of — I(g) are both positive at each critical point. As an application we show that the classical total scalar curvature A(g) = JMR dVs restricted to J4. cannot have a local minimum at any Sasakian metric.