Is there a Riemannian metric $g$ on $\mathbb{R}^2$ with inducing distance
$d$ which is not isometric to the standard metric but satisfy the property quoted bellow?



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> For every two distinct points $p,q\in \mathbb{R}^2$, the locus of all points $z$ with $d(z,p)=d(z,q)$ is a geodesic.
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As a higher dimensional version:


Is there a Riemannian metric on $\mathbb{R}^3$ which is not isometric to the standard metric but satisfy the property quoted bellow?



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> > For every two distinct points $p,q\in \mathbb{R}^3$, the locus of all points $z$ with $d(z,p)=d(z,q)$ is a 2 dimensional submanifold with constant sectional curvature?
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Edit:
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According to the comment of Willie Wong, I realized that the previous version of the question should be reconstructed. So I present the question as follows:



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> Assume that we have a complete Riemannian metric on $\mathbb{R}^n$ which satisfies the following property: For every $2$ points $p,q$, the locus of all points $\{z\mid d(z,p)=d(z,q)\}$ is a codimension- $1$ smooth submanifold which is a totally geodesic submanifold. Does this imply that the metric is isometric to either Hyperbolic space or the Euclidean space?
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A possible characterization of Euclidean geometry via the curvature of the Median-submanifold
The answer to your question is in the affirmative, even in the more general setting of general (pseudo)-Riemannian manifolds.


In the Riemannian case this is a theorem due to Busemann. Consider a Riemannian manifold $(M,g)$. The set $\Sigma\_{p,q}$ consisting of all points equidistant from $p$ and $q$ on a Riemannian manifold is called the *bisector* of $\{p,q\}$. Busemann's theorem states that if $\Sigma\_{p,q}$ are totally geodesic for any $\{p,q\}$, then $(M,g)$ has constant sectional curvature.


A generalization was proven by Beem for pseudo-Riemannian manifolds. The definition of "distance" and "bisector" requires a bit of care. Both are only defined within "simple convex neighborhoods", and the bisector is only a submanifold when $p$ and $q$ is not joined by a null geodesic. It turns out that this "local" distance function and "local" bisector, for pairs of points that are time-like or space-like separated, is enough to imply Busemann's theorem.




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### References


*Busemann, H.*, The geometry of geodesics, Pure and applied Mathematics, 6. New York: Academic Press, Inc. X, 422 p. (1955). [ZBL0112.37002](https://zbmath.org/?q=an:0112.37002).


*Beem, John K.*, [**Pseudo-Riemannian manifolds with totally geodesic bisectors**](http://dx.doi.org/10.2307/2039818), Proc. Am. Math. Soc. 49, 212-215 (1975). [ZBL0301.53026](https://zbmath.org/?q=an:0301.53026).


