- 534

- 435

## Main Question or Discussion Point

Curious, is there any useful reason to translate the 4d curved Lorentzian manifold in GR to, if i read this right, either a 46 or 230 dimensional flat Euclidian space, depending whether the manifold is compact or not? (although another source listed a 39 dimensional flat embedding).

( from https://en.wikipedia.org/wiki/Nash_embedding_theorem)

The technical statement appearing in Nash's original paper is as follows: if

for all vectors

( from https://en.wikipedia.org/wiki/Nash_embedding_theorem)

The technical statement appearing in Nash's original paper is as follows: if

*M*is a given*m*-dimensional Riemannian manifold (analytic or of class*Ck*, 3 ≤*k*≤ ∞), then there exists a number*n*(with*n*≤*m*(3*m*+11)/2 if*M*is a compact manifold, or*n*≤*m*(*m*+1)(3*m*+11)/2 if*M*is a non-compact manifold) and an injective map ƒ:*M*→**R***n*(also analytic or of class*Ck*) such that for every point*p*of*M*, the derivative dƒ*p*is a linear map from the tangent space*TpM*to**R***n*which is compatible with the given inner product on*TpM*and the standard dot product of**R***n*in the following sense:for all vectors

*u*,*v*in*TpM*. This is an undetermined system of partial differential equations