  By using a coupling method, an explicit log-Harnack inequality with local
geometry quantities is established for (sub-Markovian) diffusion semigroups on
a Riemannian manifold (possibly with boundary). This inequality as well as the
consequent $L^2$-gradient inequality, are proved to be equivalent to the
pointwise curvature lower bound condition together with the convexity or
absence of the boundary. Some applications of the log-Harnack inequality are
also introduced.
