On the Global Isometric Embedding of Pseudo-Riemannian Manifolds

Details On the Global Isometric Embedding of Pseudo-Riemannian Manifolds On the Global Isometric Embedding of Pseudo-Riemannian Manifolds

Jan 1970

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1970rspsa.314..417c&link_type=abstract

Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Volume 314, Issue 1518, pp. 417-428

Physics

40

Scientific paper

It is shown that any pseudo-Riemannian manifold has (in Nash's sense) a proper isometric embedding into a pseudo-Euclidean space, which can be made to be of arbitrarily high differentiability. The application of this to the positive definite case treated by Nash gives a new proof using a Euclidean space of substantially lower dimension. The general result is applied to the space-time of relativity, and the dimensions and signatures of the spaces needed to embed various cases are evaluated.

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