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Definition of Homogeneous Riemannian manifold: Pick a point $p \in M$. For all $q \in M$, there exists $\phi \in ISO(M)$ such that $\phi(p) = q$.

Take Riemannian metric $d$. Then, for arbitrarily small $\delta > 0$, can I assert that $B(p, \delta)$ is mapped isometrically to $B(q, \delta)$ for every $q \in M$?

James C
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    Hint: the elements of the isomotry group are isomotries of metric spaces, in that for any $\phi\in\text{ISO}(M)$, $d(x,y)=d(\phi(x),\phi(y))$. – Kajelad Apr 24 '21 at 19:10

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