Campbell's theorem (geometry)

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


Campbell's theorem, also known as Campbell’s embedding theorem and the Campbell-Magaarrd theorem, is a mathematical theorem that evaluates the asymptotic distribution of random impulses acting with a determined intensity on a damped system. The theorem guarantees 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]

References

  1. ↑ Romero, Carlos, Reza Tavakol, and Roustam Zalaltedinov. The Embedding of General Relativity in Five Dimensions. N.p.: Springer Netherlands, 2005.
  2. ↑ Lindsey, James E., et al. "On Applications of Campbell's Embedding Theorem." On Applications of Campbell's Embedding Theorem 14 (1997): 1 17. Abstract.
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