Campbell's theorem (geometry)

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 -dimensional Ricci-flat Riemannian manifold.

Statement

Campbell's theorem states that any n-dimensional Riemannian manifold can be embedded locally in an -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/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.