A Cut-Locus Characterization of the Metric Circle among Finite Metric Graphs

Abstract

We characterize metric circles among nite metric graphs by a condition on the multiplicity of minimizing geodesics. A nite metric graph containing a simple closed curve is a metric circle if and only if it admits a surjective map B such that Cut(x) ⊆ {B(x)} for every point x, where Cut(x) denotes the set of points joined to x by at least two minimizing geodesics with distinct images. Equivalently, every point of the graph occurs as a singleton cut locus. The proof is elementary: every simple closed curve meets the cut locus of every source point. As an application, a nite metric graph with a unique diametral antipode A(x) at every point and Cut(x) ⊆ {A(x)} is isometric to the metric circle of circumference twice its diameter. Examples illustrate the necessity of the cycle assumption and the limitations of antipodality conditions, showing that unique diametral antipodes, even when the antipodal map is an isometry, do not suce.

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BibTeX RIS

BibTeX

@misc{couturier2026cutlocus,
  title = {{A Cut-Locus Characterization of the Metric Circle among Finite Metric Graphs}},
  author = {Michel Le Couturier},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  url = {https://hal.science/hal-05756318v1},
  note = {CC BY-NC 4.0},
}

RIS

TY  - UNPB
TI  - A Cut-Locus Characterization of the Metric Circle among Finite Metric Graphs
AU  - Michel Le Couturier
PY  - 2026
DA  - 2026/09/21
PB  - HAL
UR  - https://hal.science/hal-05756318v1
AB  - We characterize metric circles among nite metric graphs by a condition on the multiplicity of minimizing geodesics. A nite metric graph containing a simple closed curve is a metric circle if and only if it admits a surjective map B such that Cut(x) ⊆ {B(x)} for every point x, where Cut(x) denotes the set of points joined to x by at least two minimizing geodesics with distinct images. Equivalently, every point of the graph occurs as a singleton cut locus. The proof is elementary: every simple closed curve meets the cut locus of every source point. As an application, a nite metric graph with a unique diametral antipode A(x) at every point and Cut(x) ⊆ {A(x)} is isometric to the metric circle of circumference twice its diameter. Examples illustrate the necessity of the cycle assumption and the limitations of antipodality conditions, showing that unique diametral antipodes, even when the antipodal map is an isometry, do not suce.
ER  -

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