I'd like to analyse the following problem. Suppose you are given a space (as called in relativity) or more accurately a (semi-) Riemannian manifold (in general in any dimension) that is supposed to be embedded in a space of a higher dimensions. For example the following 4-sphere,
$\begin{equation} ds^2=\sin^2\theta \:d\phi^2+d\theta^2+\cos^2\theta\:d\Omega_2^2,\:\:\:d\Omega_2^2=d\alpha^2+\sin^2\alpha \:d\beta^2, \end{equation}$
(Whether or not the coordinates cover the entire space I don't think is relevant.) and then we ask what is the form of the parametrization. That is, for the mentioned example, the 5-dimensional vector $\vec{X}(\phi,\theta,\Omega_2)$ that defines the hypersurface.
By definition $g_{\mu\nu}=\partial_\mu\vec{X}\cdot \partial_\nu\vec{X}$, with $\mu,\nu=\phi,\theta,\Omega_2$, so what we have is a set of equation that should suffice to determine $\vec{X}$, for example
$\begin{equation} \partial_\phi\vec{X}\cdot \partial_\phi\vec{X}=\sin^2\theta,\:\: \partial_\theta\vec{X}\cdot \partial_\theta\vec{X}=1, \end{equation}$
and so on. $\vec{X}$ is by no means unique, but I'd like to find a general procedure to integrate for at least one form of $\vec{X}$.
Thanks in advance.