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 $\mathbb R^N$.
$\blacksquare$
Proof
See this. It uses exterior differential systems.
$\blacksquare$
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 $\mathbb R^3$.
To find the isometric embedding $\varphi: M\to \mathbb R^N$ 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 $g=g_{ij}dx^i\otimes dx^j$ then $u$ is an isometric embedding if and only if
$$ g_{ij}=\dfrac{\partial \varphi}{\partial x^i}\cdot \dfrac{\partial \varphi}{\partial x^j} $$For example, in the case $n=2$ we have, for $U\equiv (x,u)$ an open subset of $\mathbb R^2$ and a metric $g$:
$$ g_{11}=\varphi_{1,x}^2+\varphi_{2,x}^2+\varphi_{3,x}^2 $$ $$ g_{12}=\varphi_{1,x}\varphi_{1,u}+\varphi_{2,x}\varphi_{2,u}+\varphi_{3,x}\varphi_{3,u} $$ $$ g_{22}=\varphi_{1,u}^2+\varphi_{2,u}^2+\varphi_{3,u}^2 $$________________________________________
________________________________________
________________________________________
Author of the notes: Antonio J. Pan-Collantes
INDEX: