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 $ngeq 4$ with boundary $partial M,$ nonnegative sectional curvature and zero radial Weyl tensor is either isometric, up to scaling, to the standard hemisphere $Bbb{S}^n_+,$ 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.