Restriction of Laplace operator on one-forms: from ${\mathbb{R}}^{n+2}$ and ${\mathbb{R}}^{n+1}$ ambient spaces to embedded (A)dS${}_n$ submanifolds

Abstract

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 Weitzenböck formula, for which a proof is also given.

Presentation overhauled in a definition-theorem-proof manner. A proof of Weitzenböck formula added, extending our results from Laplace-de Rham operators to Laplace-Beltrami operators. Close to published version