  In this paper, we prove a Laplacian comparison theorem for non-symmetric
diffusion operator on complete smooth $n$-dimensional Riemannian manifold
having a lower bound of modified $m$-Bakry-\'Emery Ricci tensor under $m\leq 1$
in terms of vector fields. As consequences, we give the optimal conditions for
modified $m$-Bakry-\'Emery Ricci tensor under $m\leq1$ such that the (weighted)
Myers' theorem, Bishop-Gromov volume comparison theorem, Ambrose-Myers'
theorem, Cheng's maximal diameter theorem, and the Cheeger-Gromoll type
splitting theorem hold. Some of these results were well-studied for
$m$-Bakry-\'Emery Ricci curvature under $m\geq n$ if the vector field is a
gradient type. When $m<1$, our results are new in the literature.
