On the structure of linearization of the scalar curvature

Abstract

For a compact $n$-dimensional manifold a critical point metric of the total scalar curvature functional satisfies the critical point equation (1) below, if the functional is restricted to the space of constant scalar curvature metrics of unit volume. The right-hand side in this equation is nothing but the adjoint operator of the linearization of the total scalar curvature acting on functions. The structure of the kernel space of the adjoint operator plays an important role in the geometry of the underlying manifold. In this paper, we study some geometric structure of a given manifold when the kernel space of the adjoint operator is nontrivial. As an application, we show that if there are two distinct solutions satisfying the critical point equation mentioned above, then the metric should be Einstein. This generalizes a main result in [6] to arbitrary dimension.