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The Question

How does one calculate the minimum number of dimensions of an extrinsic space that can be used to define the metric tensor

\begin{align} g_{mn} = \dfrac{\partial y^k}{\partial x^m} \dfrac{\partial y_k}{\partial x^n} \end{align}

where $y^k = (y^1 , y^2 , y^3 , \dots y^{m-2} , y^{m-1} , y^{m})^k$ is the extrinsic space of $m$ dimensions and $x^k = (x^1 , x^2 , x^3 , \dots x^{m-2} , x^{m-1} , x^{n})^k$ is the embedded space of $n$ dimensions.

Related Question

This questions extends from a previous question: How to prove the covariant derivative cannot be written as an eigendecomposition of the partial derivative?.

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    $\begingroup$ Are you asking for a proof of the Nash embedding theorem? If that is the case, maybe math.stackexchange.com would be a better place? $\endgroup$ – Danu Mar 9 '14 at 18:14
  • $\begingroup$ It should be said that nash only gives a worse case and you can very often do a lot better. $\endgroup$ – Jerry Schirmer Mar 9 '14 at 22:40

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