  We obtain a Bonnet-Myers theorem under a spectral condition: a closed
Riemannian manifold $(M^n,g)$ for which the lowest eigenvalue of the Ricci
tensor $\rho$ is such that the Schr\"odinger operator $(n-2)\Delta + \rho$ is
positive has finite fundamental group. As a continuation of our earlier
results, we obtain isoperimetric inequalities from a Kato condition on the
Ricci curvature. Furthermore, we obtain the Kato condition for the Ricci
curvature under purely geometric assumptions.
