Consider the case of the 4-dimensional de Sitter space, $dS_4$: the hyperboloid given by $$-x_0^2+x_1^2+x_2^2+x_3^2+x_4^2=\alpha^2,$$
embedded in in 5-dimensional Minkowski spacetime, $R^{1,4}$, with metric $$ds^2_M=-dx_0^2+dx_1^2+dx_2^2+dx_3^2+dx_4^2.$$
It is known (and easy to check) that in the coordinates $(t,y_1,y_2,y_3)$
given by
- $x_0= \alpha \sinh(t/\alpha)+r^2 e^{t / \alpha}/ 2 \alpha=f_0(t,y_1,y_2,y_3)$
- $x_1= \alpha \cosh(t/\alpha)-r^2 e^{t / \alpha}/ 2 \alpha=f_1(t,y_1,y_2,y_3)$
- $x_i= e^{t/ \alpha} y_{i-1}=f_i(t,y_1,y_2,y_3)$
for $r^2=y_1^2+y_2^2+y_3^2$, and $i=2,3,4$,
the induced metric on the hyperboloid is $$ds^2_{\text{dS}}=-dt^2+e^{2 t/{ \alpha}}(dy_1^2+dy_2^2+dy_3^2).$$
Now, the question is the following : Let assume that we do not know the functions $f_0,f_1,...f_4$ but we consider the ansatz
$$ds^2_{\text{dS}}=-dt^2+e^{2 A(t)}(dy_1^2+dy_2^2+dy_3^2).$$
Is there a systematic way of finding the functions $A(t),f_0(t),f_1(t),...,f_4(t)$?
Of course, the first idea is to proceed by writing the the metric involving the unknown functions, setting it equal to the desired form, and along with the hyperboloid equation, find the desired functions. However, this seems to be a highly nontrivial problem. Is there any theory behind this that makes the task simpler?