The Holonomy of Optimal Mass Transport: The Smooth Case

Mahmoud Abdelgalil, Tryphon T. Georgiou

arXiv:2608.15585v1 Primary: math.DG Secondary: math.OC

Abstract

We prove that, on a smooth $n$-dimensional Riemannian manifold without boundary, any vector field can be written as a linear combination of, at most, $\max\{6,6n-3\}$ depth one Lie brackets of pairs of gradient vector fields. Utilizing this along with the strong Trotter property, we show that, if the manifold is also compact and connected, the group generated by diffeomorphic optimal transport maps is dense in the identity component of the diffeomorphism group.

Disclosure

“d, after minor changes, to pseudo-Riemannian manifolds since the key identity (12) is valid for any non-degenerate symmetric bilinear form. Disclosure The absorption identity (12), key to our main result, was produced in interaction with Claude Fable 5 (Anthropic, 2026), and seems to be novel. The authors have verified the mathematical details and are solely responsible for the content. 2”
PDF page 2

Classification: Substantial mathematical content or result generation
Multiplier: 10
Verified: 2026-08-23T05:17:39.459280Z

Structural counts

Pages 6 pdf

Theorems 2 source

Lemmas 0 source

Propositions 2 source

Corollaries 0 source

Definitions 0 source

Displayed equations 33 source

Bibliography entries 21 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file main.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.