Massive scalar field on (A)dS space from a massless conformal field in ${\mathbb{R}}^6$

We show how the equations for the scalar field (including the massive, massless, minimally and conformally coupled cases) on de Sitter and Anti-de Sitter spaces can be obtained from both the SO$(2,4)$-invariant equation $\mathrm{\square }\varphi =0$ in ${\mathbb{R}}^6$ and two geometrical constraints defining the (A)dS space. Apart from the equation in ${\mathbb{R}}^6$, the results only follow from the geometry.