Let me ask this question without too much formalization:
Suppose a smooth surface $M$ has the property that for all spheres $S(p,R)$ (i.e. the set of all points which lie a distance $R\geq 0$ from $p \in M$, with distance as measured inside the surface), there is always a different point $q(p,R) \in M$ and a distance $D(p,R)\geq 0$ so that $$S(p,R)=S(q(p,R),D(p,R))$$ In words: both $p$ and $q(p,R) \neq p$ are a center of the sphere $S(p,R)$.
Q: Is $M$ (a part of) a sphere? More formally: is $M$ isometric to (a subset of) a sphere?
Remark: Spheres clearly possess the mentioned property, with every sub-sphere having one point $p$ ánd its antipodal $\pi(p)$ at its center. This property is maintained when we remove a finite number of pairs of antipodal points from such a sphere or when we take a few disconnected spheres.
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