AdS space as an example of klein geometry

Question

If we start with the definition of the coset $AdS_{D+2}:=\frac{O(2,D)}{O(1,D)}$ , How do we derive the constraint equation for the AdS coordinates $\mu \nu -(X^{i})^{2}=R^{2}$ ?

Answer 1

Consider a vector in $\mathbb{R}^{2,D}$ with norm $-R^{2}$ , The set of all vectors with this norm are rotated into each other by the $O(2,D)$ Rotations. Use this group to make the vector in the form $X=(1,0,0...)$ It is obvious that the isotropy group that leaves this invariant is $O(1,D)$ and thus we get the equivalence because these vectors with the specified norm are in one to one correspondance with the group transformations modulo the isotropy group.