  In this paper we obtain generalized Calabi-type compactness criteria for
complete Riemannian manifolds that allow the presence of negative amounts of
Ricci curvature. These, in turn, can be rephrased as new conditions for the
positivity, for the existence of a first zero and for the
nonoscillatory-oscillatory behaviour of a solution $g(t)$ of $g"+Kg=0$,
subjected to the initial condition $g(0)=0$, $g'(0)=1$. A unified approach for
this ODE, based on the notion of critical curve, is presented. With the aid of
suitable examples, we show that our new criteria are sharp and, even for $K\ge
0$, in borderline cases they improve on previous works of Calabi, Hille-Nehari
and Moore.
