Rigidity of complete manifolds with weighted Poincaré inequality

We consider complete Riemannian manifolds which satisfy a weighted Poincarè inequality and have the Ricci curvature bounded below in terms of the weight function. When the weight function has a non-zero limit at infinity, the structure of this class of manifolds at infinity are studied and certain splitting result is obtained. Our result can be viewed as an improvement of Li-Wang’s result in \cite{LW3}.