Cartan-Janet theorem

Theorem
Let $(M, g)$ be a real-analytic Riemannian manifold of dimension $n$ , and let $N = \frac{1}{2} n (n + 1)$ . Every point of $M$ has a neighbourhood which has a real-analytic isometric embedding into $R^{N}$ .
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Proof
See this. It uses exterior differential systems.
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Keep an eye: the smooth case is still open even in dimension 2.

Specifically, for the case $n = 2$ it means that locally any Riemannian surface can be isometrically embedded in $R^{3}$ .

To find the isometric embedding $φ : M \to R^{N}$ we have to solve a system of fully non-linear first-order PDEs for $u$ . Namely, if the coordinates of $M$ are $(x^{i})$ and the metric is $g = g_{i j} d x^{i} \otimes d x^{j}$ then $u$ is an isometric embedding if and only if

g_{i j} = \frac{\partial φ}{\partial x^{i}} \cdot \frac{\partial φ}{\partial x^{j}}

For example, in the case $n = 2$ we have, for $U \equiv (x, u)$ an open subset of $R^{2}$ and a metric $g$ :

g_{11} = φ_{1, x}^{2} + φ_{2, x}^{2} + φ_{3, x}^{2}

g_{12} = φ_{1, x} φ_{1, u} + φ_{2, x} φ_{2, u} + φ_{3, x} φ_{3, u}

g_{22} = φ_{1, u}^{2} + φ_{2, u}^{2} + φ_{3, u}^{2}