  We consider the first eigenvalue $\lambda_1(\Omega,\sigma)$ of the Laplacian
with Robin boundary conditions on a compact Riemannian manifold $\Omega$ with
smooth boundary, $\sigma\in\bf R$ being the Robin boundary parameter. When
$\sigma>0$ we give a positive, sharp lower bound of $\lambda_1(\Omega,\sigma)$
in terms of an associated one-dimensional problem depending on the geometry
through a lower bound of the Ricci curvature of $\Omega$, a lower bound of the
mean curvature of $\partial\Omega$ and the inradius. When the boundary
parameter is negative, the lower bound becomes an upper bound. In particular,
explicit bounds for mean-convex Euclidean domains are obtained, which improve
known estimates.
  Then, we extend a monotonicity result for $\lambda_1(\Omega,\sigma)$ obtained
in Euclidean space by Giorgi and Smits to a class of manifolds of revolution
which include all space forms of constant sectional curvature. As an
application, we prove that $\lambda_1(\Omega,\sigma)$ is uniformly bounded
below by $\frac{(n-1)^2}4$ for all bounded domains in the hyperbolic space of
dimension $n$, provided that the boundary parameter $\sigma\geq\frac{n-1}{2}$
(McKean-type inequality). Asymptotics for large hyperbolic balls are also
discussed
