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

The FLRW spacetimes can be realized as submanifolds of ${\mathbb{R}}^6$. In this paper we relate the Laplace-Beltrami operator for an homogeneous scalar field $\varphi$ of ${\mathbb{R}}^6$ to its explicit restriction on FLRW spacetimes. We then make the link between the homogeneous solutions of the equation ${\mathrm{\square }}_6\varphi =0$ in ${\mathbb{R}}^6$ and those of the Klein-Gordon equation $({\mathrm{\square }}_f-\xi R^f+m^2){\varphi }^f=0$ for the free field ${\varphi }^f$ in the FLRW spacetime. We obtain as a byproduct a formula for the Ricci scalar of the FRLW spacetime in terms of the function $f$ defining this spacetime in ${\mathbb{R}}^6$.