  On a fairly general class of Riemannian manifolds M, we prove lower estimates
in terms of the Ricci curvature for the spectral bound (when M has infinite
volume) and for the spectral gap (when M has finite volume) for the
Laplace-Beltrami operator. As a byproduct of our results we obtain an extension
of the Bonnet-Myers theorem on the compactness of the manifold. We also prove
lower bounds for the spectral gap for Ornstein-Uhlenbeck type operators on
weighted manifolds. As an application we prove lower bounds for the spectral
gap of perturbations of some radial measures on R n .
