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