  We consider complete Riemannian manifolds which satisfy a weighted Poincar\`e
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}.
