Let $M$ be a 2-dimensional Riemannian manifold, $x,y \in M$. Why is the set of points $\{z | d(z,x) = d(z,y)\}$ a geodesic?
What can we say about higher-dimensional Riemannian manifolds?
2026-03-27 01:47:47.1774576067
Why is the geometric locus of points equidistant to two other points in a two-dimensional Riemannian manifold a geodesic?
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John K. Beem, "Pseudo-Riemannian Manifolds with Totally Geodesic Bisectors" Proceedings of the American Mathematical Society Vol. 49, No. 1 (1975), pp. 212-215.
It follows from the main result of this paper that if $M$ is a Riemannian manifold such that the bisector of every pair of points is a totally geodesic submanifold, then $M$ has constant sectional curvature. This theorem is due to Busemann, but the paper is more accessible.
The converse is also true, which is a pleasant and elementary calculation.