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

TL;DR

This paper investigates how the Laplace-de Rham operator on one-forms behaves when restricted from ambient spaces like b^{n+2} and b^{n+1} to embedded pseudo-spheres, including (A)dS spaces, providing explicit formulas and extension results.

Contribution

It derives explicit formulas for restricting and extending the Laplace-de Rham operator on one-forms between ambient spaces and embedded (A)dS submanifolds, including the case of additional terms.

Findings

Explicit restriction formulas for Laplace-de Rham operator on (A)dS spaces.

Extension results showing ambient Laplace-de Rham operator restrictions.

Translation of results to Laplace-Beltrami operator via Weitzenb4ck formula.

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 Weitzenb\"ock formula, for which a proof is also given.