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 $phi$ 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 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 $mathbb{R}^6$.