Restriction of Laplace operator on one-forms: From Rn+2 and Rn+1 ambient spaces to embedded (A)dSn submanifolds

Abstract

The Laplace–de Rham operator acting on a one-form a: □a in Rn+2 or Rn+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 the Laplace–de Rham operator on ambient spaces is also studied. We show that for any additional term, this operator on the embedded space is the restriction of the 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.