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The FLRW spacetimes can be realized as submanifolds of $\mathbb{R}^6$. Inthis paper we relate the Laplace-Beltrami operator for an homogeneous scalarfield $\phi$ of $\mathbb{R}^6$ to its explicit restriction on FLRW spacetimes.We then make the link between the homogeneous solutions of the equation$\square_6 \phi = 0$ in $\mathbb{R}^6$ and those of the Klein-Gordon equation$(\square_{f} - \xi R^f + m^2)\phi^f=0 $ for the free field $\phi^f$ in theFLRW spacetime. We obtain as a byproduct a formula for the Ricci scalar of theFRLW spacetime in terms of the function $f$ defining this spacetime in$\mathbb{R}^6$.

FLRW spaces as submanifolds of $\mathbb{R}^6$: restriction to the Klein-Gordon operator