A $D$-dimensional de Sitter spacetime $dS_D$ is a timelike hyperbola embedded in $(D+1)$-dimensional Minkowski spacetime $\mathfrak M_{D+1}$. The metric in the embedding space is
$$ ds^2 = (dX_0)^2 - \sum_{A=1}^D (dX_A)^2 \,.$$
With a radius $l\gt0$, $dS_D$ is defined by
$$ -\eta_{AB}X^AX^B = l^2 \,. $$
I would like to prove the popular claim that
$$ \boxed{ dS_D = \frac{O(1,D)}{O(1,D-1)} = \frac{SO(1,D)}{SO(1,D-1)} }\,.$$
Note that $O(1,D)$ is the Lorentz group which is a part of the isometry group of $\mathfrak M_{D+1}$ and $ O(1,D-1) \subset O(1,D) $ is the stabilizer of $dS_D$.
Alternatively, one can see that under a Wick rotation $X_0 \mapsto iX_0,\ dS_D \mapsto S^D \subset \mathbb R^{D+1}\,.$ So, perhaps an equivalent claim would be the following.
$$ \boxed{ S^D = \frac{O(D+1)}{O(D)} = \frac{SO(D+1)}{SO(D)} }\,.$$
A question requesting a proof of the above has been made already, but no answers have been received so far.
Another example is the following. A $D$-dimensional anti de Sitter spacetime $AdS_D$ is embedded in a $(D+1)$-dimensional spacetime with a Lorentzian metric of index 2.
$$ ds^2 = (dX_0)^2 - \sum_{A=1}^{D-1} (dX_A)^2 + (dX_D)^2 \,.$$
With a radius $l\gt0$, $AdS_D$ is defined by
$$ \eta_{AB}X^AX^B = l^2 \,. $$
Now the claim is the following:
$$ \boxed { AdS_D = \frac{O(2,D-1)}{O(1,D-1)} = \frac{SO(2,D-1)}{SO(1,D-1)} }\,.$$
Note that each of the boxed claims are (loosely speaking) equivalent modulo various Wick rotations. I would appreciate even a sketch of proof for above claims.
Hints:
(1) First prove that for every $v\in dS_D$ there exists $g\in O(1,D)$ such that $g(v)={\mathbf e}_1= (0,1,0,0,...0)$.
(2) Then check that the stabilizer of ${\mathbf e}_1$ in $O(1,D)$ is isomorphic to $O(1,D-1)$.
(3) Next, define a continuous (actually, smooth) map $O(1,D)\to dS_D$ sending $g\in O(1,D)$ to $g({\mathbf e}_1)$.
(4) Check that this map descends to a continuous (actually, smooth) injective map $f: O(1, D)/O(1,D-1)\to ds_D$ (using Part 2).
(5) Check (using Part 1) that $f$ is surjective.
Thus, you have a bijective continuous map between two topological manifolds, which implies that $f$ is a homeomorphism.
(6) If you still care about the problem after the first 5 steps, show that $f$ is a diffeomorphism by appealing, for instance, to the Constant Rank Theorem.