  The Laplace-de Rham operator acting on a one-form $a$: $\square a$, in
$\mathbb{R}^{n+2}$ or $\mathbb{R}^{n+1}$ spaces is restricted to
$n$-dimensional pseudo-spheres. This includes, in particular, the
$n$-dimensional de Sitter and Anti-de Sitter space-times. The restriction is
designed to extract the corresponding $n$-dimensional Laplace-de Rham operator
acting on the corresponding $n$-dimensional one-form on pseudo-spheres.
Explicit formulas relating these two operators are given in each situation. The
converse problem, of extending an $n$-dimensional operator composed of the sum
of the Laplace-de Rham operator and additional terms to ambient spaces
Laplace-de Rham operator, is also studied. We show that for any additional term
this operator on the embedded space is the restriction of Laplace-de Rham
operator on the embedding space.These results are translated to the
Laplace-Beltrami operator thanks to the Weitzenb\"ock formula, for which a
proof is also given.
