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The problem of cosmological graviton creation for homogeneous and isotropicuniverses with elliptical ($\vae =+1$) and hyperbolical ($\vae =-1$) geometriesis addressed. The gravitational wave equation is established for aself-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 interms of hypergeometric functions for any value of $\gamma$ and spatialcurvature $\vae$. An expression representing an adiabatic vacuum state is thenobtained in terms of associated Legendre functions whenever $\gamma\neq\frac{2}{3}\; \frac{(2n+1)}{(2n-1)}$, where n is an integer. This includes mostcases of physical interest such as $\gamma =0,\;4/3\;,1$. The mechanism ofgraviton creation is reviewed and the Bogoliubov coefficients related totransitions between arbitrary cosmic eras are also explicitly evaluated.Comment: 25 pages, uses REVTE

Graviton production in elliptical and hyperbolic universes