The Reeb Transform
Abstract
Reeb Transforms offer a compact representation of how shapes in ℝn change when sliced along varying directions. We define the Reeb Transform as the family of Reeb graphs induced by height functions along all directions in the unit sphere. In this paper, we develop a rigorous treatment of Reeb Transforms for o-minimal definable sets, with particular emphasis on one-dimensional stratified spaces embedded in ℝd and on surfaces in ℝ3. We establish injectivity of the Reeb Transform in multiple settings, including compact surfaces in ℝ3, capturing the essential topological features needed to uniquely reconstruct such surfaces. However, in dimensions above three, the Reeb Transform ceases to be injective, indicating the limitations of this descriptor in higher-dimensional settings.
Related
- Algebraic realization of stable Poincaré-Reeb graphs
- Existence of Reeb Chords from the Generation-Criteria
- Multiplicity of brake Reeb orbits on real prequantization bundles
- On Reeb spaces of non-proper smooth functions which are $1$-dimensional CW complexes
- Contractible Closed Reeb Orbits on Prequantization Bundles under Weak Index Conditions
BibTeX
@misc{chambers2026reeb,
title = {{The Reeb Transform}},
author = {Erin Chambers and Shankha Shubhra Mukherjee and Katharine Turner},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03126},
eprint = {2610.03126},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03126},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Reeb Transform
AU - Erin Chambers
AU - Shankha Shubhra Mukherjee
AU - Katharine Turner
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03126
UR - https://arxiv.org/abs/2610.03126
AB - Reeb Transforms offer a compact representation of how shapes in ℝ^n change when sliced along varying directions. We define the Reeb Transform as the family of Reeb graphs induced by height functions along all directions in the unit sphere. In this paper, we develop a rigorous treatment of Reeb Transforms for o-minimal definable sets, with particular emphasis on one-dimensional stratified spaces embedded in ℝ^d and on surfaces in ℝ^3. We establish injectivity of the Reeb Transform in multiple settings, including compact surfaces in ℝ^3, capturing the essential topological features needed to uniquely reconstruct such surfaces. However, in dimensions above three, the Reeb Transform ceases to be injective, indicating the limitations of this descriptor in higher-dimensional settings.
ER -