Remarks on compact quasi-Einstein manifolds with boundary

In this paper, we prove that a compact quasi-Einstein manifold $(M^n,g,u)$ of dimension $n\ge 4$ with boundary $\mathrm{\partial }M,$ nonnegative sectional curvature and zero radial Weyl tensor is either isometric, up to scaling, to the standard hemisphere ${\mathbb{S}}_{+}^n,$ or $g=dt^{2}+\psi ^{2}(t)g_{L}$and$u=u(t),$where$g_{L}$ is Einstein with nonnegative Ricci curvature. A similar classification result is obtained by assuming a fourth-order vanishing condition on the Weyl tensor. Moreover, a new example is presented in order to justify our assumptions. In addition, the case of dimension $n=3$ is also discussed.