  On closed Riemannian manifolds with Bakry-\'Emery Ricci curvature bounded
from below and bounded gradient of the potential function, we obtain lower
bounds for all positive eigenvalues of the Beltrami-Laplacian instead of the
drifted Laplacian. The lower bound of the $k$th eigenvalue depends on $k$,
Bakry-\'Emery Ricci curvature lower bound, the gradient bound of the potential
function, and the dimension and diameter upper bound of the manifold, but the
volume of the manifold is not involved. Especially, these results apply to
closed manifolds with Ricci curvature bounded from below.
