Campbell's theorem (geometry)

This article is about the embedding of Riemannian manifolds. For other uses, see Campbell's theorem (probability).

Campbell's theorem, named after John Edward Campbell, also known as Campbell’s embedding theorem and the Campbell-Magaard theorem, is a mathematical theorem guaranteeing that any n-dimensional Riemannian manifold can be locally embedded in an (n + 1)-dimensional Ricci-flat Riemannian manifold.^[1]

Statement

Campbell's theorem states that any n-dimensional Riemannian manifold can be embedded locally in an (n + 1)-manifold with a Ricci curvature of R_'a b = 0. The theorem also states, in similar form, that an n-dimensional pseudo-Riemannian manifold can be both locally and isometrically embedded in an n(n + 1)/2-pseudo-Euclidean space.

Applications

Campbell’s theorem can be used to produce the embedding of numerous 4-dimensional spacetimes in 5-dimensional Ricci-flat spaces. It is also used to embed a class of n-dimensional Einstein spaces.^[2]