Let $$\mathcal {C}$$ C be the space of smooth metrics g on a given compact manifold $$M^{n}$$ M n ( $$n\ge 3$$ 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 $$\mathcal {C}$$ 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, $$4\le n\le 10,$$ 4 ≤ n ≤ 10 , CPE metric satisfying a $$L^{n/2}$$ 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.