On critical point equation of compact manifolds with zero radial Weyl curvature

Let C be the space of smooth metrics g on a given compact manifold M-n (n >= 3) with constant scalar curvature and unitary volume. The goal of this paper is to study the critical point of the total scalar curvature functional restricted to the space C (we shall refer to this critical point as CPE metrics) under assumption that (M, g) has zero radial Weyl curvature. Among the results obtained, we emphasize that in 3-dimension we will be able to prove that a CPE metric with nonnegative sectional curvature must be isometric to a standard 3-sphere. We will also prove that a n-dimensional, 4 <= n <= 10, CPE metric satisfying a L-n/2-pinching condition will be isometric to a standard sphere. In addition, we shall conclude that such critical metrics are isometrics to a standard sphere under fourth-order vanishing condition on the Weyl tensor.