  The problem of cosmological graviton creation for homogeneous and isotropic
universes with elliptical ($\vae =+1$) and hyperbolical ($\vae =-1$) geometries
is addressed. The gravitational wave equation is established for a
self-gravitating fluid satisfying the barotropic equation of state $p=(\gamma
-1)\rho$, which is the source of the Einstein's equations plus a cosmological
$\Lambda$-term. The time dependent part of this equation is exactly solved in
terms of hypergeometric functions for any value of $\gamma$ and spatial
curvature $\vae$. An expression representing an adiabatic vacuum state is then
obtained in terms of associated Legendre functions whenever $\gamma\neq
\frac{2}{3}\; \frac{(2n+1)}{(2n-1)}$, where n is an integer. This includes most
cases of physical interest such as $\gamma =0,\;4/3\;,1$. The mechanism of
graviton creation is reviewed and the Bogoliubov coefficients related to
transitions between arbitrary cosmic eras are also explicitly evaluated.
