Harmonic maps and isometric embeddings of the spacetime

Abstract

We investigate harmonic maps in the context of isometric embeddings when the target space is Ricci-flat and has codimension one. With the help of the Campbell-Magaard theorem we show that any $n$-dimensional ($n⩾3$) Lorentzian manifold can be isometrically and harmonically embedded in a (n+1)-dimensional semi-Riemannian Ricci-flat space. We then extend our analysis to the case when the target space is an Einstein space. Finally, as an example, we work out the harmonic and isometric embedding of a Friedmann-Robertson-Walker spacetime in a five-dimensional Ricci-flat space and proceed to obtain a general scheme to minimally embed any vacuum solution of general relativity in Ricci-flat spaces with codimension one.

17 pages/ RevTex

References in corpus (5)

• Application of the Campbell-Magaard theorem to higher-dimensional physics
• The embedding of the spacetime in five dimensions: an extension of Campbell-Magaard theorem
• Embeddings in Non-Vacuum Spacetimes
• The embedding of the spacetime in five-dimensional spaces with arbitrary non-degenerate Ricci tensor
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Cited by in corpus (5)

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• On the embedding of Weyl manifolds