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.
In this article, we study four-dimensional complete gradient shrinking Ricci solitons. We prove that a four-dimensional complete gradient shrinking Ricci soliton satisfying a pointwise condition involving either the self-dual or anti-self-dual part o
f the Weyl tensor is either Einstein, or a finite quotient of either the Gaussian shrinking soliton $Bbb{R}^4,$ or $Bbb{S}^{3}timesBbb{R}$, or $Bbb{S}^{2}timesBbb{R}^{2}.$ In addition, we provide some curvature estimates for four-dimensional complete gradient Ricci solitons assuming that its scalar curvature is suitable bounded by the potential function.
The goal of this article is to investigate nontrivial $m$-quasi-Einstein manifolds globally conformal to an $n$-dimensional Euclidean space. By considering such manifolds, whose conformal factors and potential functions are invariant under the action
of an $(n-1)$-dimensional translation group, we provide a complete classification when $lambda=0$ and $mgeq 1$ or $m=2-n.$
