Given any Riemannian or Semi-Riemannian manifold $(M,g)$, does there exist a Eucildean space $(E,g^\prime)$ of enough high dimension with metric $g^\prime=diag\{-1,-1,...,+1,+1,...\}$ with any n copies of "$-1$" and m "$+1$" such that $(M,g)$ is embedded in $(E,g^\prime)$ and $g$ is the induced metric of $g^\prime$ ?
$\begingroup$
$\endgroup$
1
-
7$\begingroup$ For Riemannian manifolds this is the Nash embedding theorem: en.wikipedia.org/wiki/Nash_embedding_theorem $\endgroup$– Lennart MeierCommented Nov 1, 2014 at 15:48
Add a comment
|
2 Answers
$\begingroup$
$\endgroup$
Yes, have a look at Robert Greene's book Isometric Embeddings of Riemannian and Pseudo Riemannian Manifolds, Volume 97 of Memoirs of the American Mathematical Society Memoirs, 1970.
answered Nov 1, 2014 at 13:46
$\begingroup$
$\endgroup$
For the Lorentzian case see http://arxiv.org/abs/0812.4439. In this article the authors give conditions for the existence of the imbedding.
