Campbell's theorem (geometry)

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.

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]