Splitting theorems on complete manifolds with Bakry-\'Emery curvature

In this paper we study some splitting properties on complete noncompact manifolds with smooth measures when $\mathrm{\infty }$-dimensional Bakry-'Emery Ricci curvature is bounded from below by some negative constant and spectrum of the weighted Laplacian has a positive lower bound. These results extend the cases of Ricci curvature and $m$-dimensional Bakry-'Emery Ricci curvature.