Isotropy and homogeneity are important notions in cosmology. Isotropy at a point $p$ of the manifold $M$ means for two vectors $V,W$ in the tangent space $T_pM$, there is an isometry of $M$ that takes $V$ to be parallel to $W$. Homogeneity of the manifold means for any two points $p,q$ of $M$, there is an isometry that takes $p$ to $q$. I found in different places the statement that isotropy everywhere implies homogeneity. But how to prove this statement?
Intuitively, by isotropy we can always bring two points closer, for example, by flipping $p$ around a point $p^{'}$ between $p,q$. I am not sure if this line of thought can lead to a rigorous proof, as less technical as possible.
Edit: As pointed out by Janisch and Desteny, we can choose $r$ the mid point of the geodesic connecting the two points $p,q$. The isometry that flips the tangent vector of the geodesic at $r$ toward $p$ will take $p$ to $q$.
