Let $G$ be a Lie group, and $H$ a Lie subgroup of $G$ such that $G/H\sim \mathcal M$ is a homogeneous space diffeomorphic to $\mathbb{R}^n$, equipped with an invariant Riemannian metric $g$. If this help, $\mathcal M$ can be assumed to be a Riemannian symmetric space.
Assume that:
- there is a diffeomorphism $\pi = (\pi_A,\pi_B) : \mathcal M \to A \times B$ where $A$ and $B$ are $2$ manifolds, such that submanifolds of the type $S^A_y = \pi^{-1}(A,y)$ and $S^B_y = \pi^{-1}(x,B)$ are totally geodesic submanifolds of $\mathcal M$.
- the action of $G$ on $\mathcal M$ can be turned into a transitive action on slices $S^A_y$, and a transitive action on slices $S^B_x$
Let $o\in \mathcal M$. The metric $g$ on $\mathcal M$ induces a metric on the slice $S^{A}_{\pi_B(o)}$ which can be pushed by $\pi_A$ to a metric $g_A$ on $A$. Similarly we get a metric $g_B$ on $B$.
Can we show that $\pi$ is an isometry between $(\mathcal M,g)$ and $(A\times B,g_A \oplus g_B)$ ?
I am pretty convinced that it is true, but I struggle to put all the pieces together. Using the De Rham product decomposition theorem, I believe that we can prove that $(\mathcal M,g)$ and $(A\times B,g_A \oplus g_B)$ are isometric, but I don't see how to show that $\pi$ is an isometry.
Your help would be very appreciated.
