Friedmann-Lemaître-Robertson-Walker spaces as submanifolds of R6\mathbb{R}^6: Restriction to the Klein-Gordon operator

International audienceThe Friedmann-Lemaître-Robertson-Walker (FLRW) spacetimes can be realized as submanifolds of R6. In this paper, we relate the Laplace-Beltrami operator for a homogeneous scalar field ϕ of R6 to its explicit restriction on FLRW spacetimes. We then make the link between the homogeneous solutions of the equation □6ϕ=0 in R6 and those of the Klein-Gordon equation (□f−ξRf+m2)ϕf=0 for the free field ϕf in the FLRW spacetime. We obtain as a byproduct a formula for the Ricci scalar of the FLRW spacetime in terms of the function f defining this spacetime in R6