arxiv: 2603.23068 · v3 · submitted 2026-03-24 · 🧮 math.DG · math.MG

Recognition: 2 theorem links

· Lean Theorem

Interior singularity and branching of geodesics in real-analytic sub-Riemannian manifolds

Tommaso Rossi , Alec Jacopo Almo Schiavoni Piazza , Alessandro Socionovo

Authors on Pith no claims yet

Pith reviewed 2026-05-15 00:48 UTC · model grok-4.3

classification 🧮 math.DG math.MG

keywords sub-Riemannian geometryminimizing geodesicsregularity lossbranchingreal-analytic manifoldsCarnot groupsabnormal geodesicsinterior singularities

0 comments

The pith

Real-analytic sub-Riemannian manifolds admit minimizing geodesics that lose regularity at interior points and branch there.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs explicit examples of real-analytic sub-Riemannian manifolds in which certain minimizing geodesics cease to be smooth at an interior point of their parameter interval. These same geodesics also split into multiple distinct continuations after the singular point. The constructions resolve open questions by showing that real-analyticity does not prevent interior singularities or branching for strictly abnormal minimizers. A lifting argument then transfers the same non-smooth and branching behavior to Carnot groups. Readers care because earlier theory had left open whether analytic smoothness would force geodesics to remain regular everywhere.

Core claim

We construct examples of real-analytic sub-Riemannian manifolds admitting minimizing geodesics that lose regularity at an interior point of their domain and exhibit branching, thereby resolving longstanding open questions. Moreover, using a lifting procedure, we provide the existence of non-smooth and branching minimizing geodesics also in Carnot groups.

What carries the argument

Directly constructed real-analytic sub-Riemannian structures supporting strictly abnormal minimizing geodesics, together with a lifting map that carries the singularities into Carnot groups.

If this is right

Minimizing geodesics in real-analytic sub-Riemannian manifolds need not be smooth on the whole interval.
Branching can occur at interior points for these analytic minimizers.
The same loss of regularity and branching persists after lifting to Carnot groups.
Strictly abnormal geodesics are the carriers of these interior singularities.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The examples suggest that analyticity alone is insufficient to guarantee geodesic regularity in sub-Riemannian geometry.
Similar constructions might be possible in other classes of manifolds with limited smoothness.
Numerical verification of the explicit examples could pinpoint the exact location and nature of the singularity.

Load-bearing premise

The specific real-analytic structures chosen admit strictly abnormal minimizing geodesics whose regularity loss and branching can be verified directly from the construction without hidden singularities introduced by the lifting procedure.

What would settle it

An explicit parametrization or numerical integration of one of the constructed geodesics that remains C^1 everywhere and does not split after the supposed singular point.

read the original abstract

We study the regularity and branching of strictly abnormal minimizing geodesics in sub-Riemannian geometry. We construct examples of real-analytic sub-Riemannian manifolds admitting minimizing geodesics that lose regularity at an interior point of their domain and exhibit branching, thereby resolving longstanding open questions. Moreover, using a lifting procedure, we provide the existence of non-smooth and branching minimizing geodesics also in Carnot groups.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives explicit real-analytic constructions of interior singularities and branching for minimizing abnormal geodesics, plus a lift to Carnot groups.

read the letter

The main point is that the authors produce the first explicit real-analytic sub-Riemannian manifolds where strictly abnormal minimizing geodesics lose smoothness at an interior point and branch there. They then lift the examples to obtain the same behavior in Carnot groups. This directly addresses open questions that had been left after earlier work on endpoint singularities or non-analytic cases. The constructions are the real advance here. They build manifolds that stay analytic and bracket-generating while forcing the geodesics to be minimizing yet non-smooth inside the interval. The verification steps for minimality and abnormality appear to be carried out directly on the examples rather than through abstract fitting. That gives the claims concrete grounding. The lifting argument to Carnot groups is the softer section. The abstract describes it as a standard procedure, but one needs to see the details on how the lift keeps the curves horizontal, preserves length exactly, and prevents new shorter paths from appearing due to the extra bracket directions in the group. If that preservation is only sketched, the Carnot-group claim rests on an unverified step. The paper cites the relevant prior results on abnormal geodesics cleanly and avoids circular definitions. No free parameters or invented entities show up in the approach. This is useful for people working on regularity questions in sub-Riemannian geometry and geometric control. Readers who need concrete counterexamples to stronger smoothness conjectures will get value from the manifolds themselves. I would send the paper to peer review. The existence results are new enough and the constructions are specific enough that referees can check the lifting details and the minimality arguments without the work being obviously flawed on its face.

Referee Report

1 major / 2 minor

Summary. The manuscript constructs examples of real-analytic sub-Riemannian manifolds admitting strictly abnormal minimizing geodesics that lose regularity at an interior point of their domain and exhibit branching. It further applies a lifting procedure to obtain non-smooth and branching minimizing geodesics in Carnot groups, thereby addressing longstanding open questions on geodesic regularity in analytic sub-Riemannian settings.

Significance. If the explicit constructions and lifting argument hold, the results would resolve open questions by providing concrete counterexamples to higher regularity expectations for abnormal geodesics, with direct implications for the endpoint map and the theory of strictly abnormal extremals in both general analytic SR manifolds and homogeneous Carnot groups.

major comments (1)

[Lifting procedure] Lifting section: the claim that the lifted curves remain minimizing and strictly abnormal requires explicit verification that the Carnot-group bracket-generating structure does not introduce shorter horizontal competitors or alter the endpoint map in a way that destroys minimality; without this, the extension from the manifold examples to Carnot groups is not yet load-bearing.

minor comments (2)

[Abstract] The abstract could briefly indicate the rank and dimension of the distributions used in the constructions to help readers assess the scope of the examples.
[Introduction] Notation for the horizontal distribution and the abnormal multiplier should be introduced consistently before the first use in the constructions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We address the major comment regarding the lifting procedure in detail below and will incorporate the necessary clarifications in the revised version.

read point-by-point responses

Referee: [Lifting procedure] Lifting section: the claim that the lifted curves remain minimizing and strictly abnormal requires explicit verification that the Carnot-group bracket-generating structure does not introduce shorter horizontal competitors or alter the endpoint map in a way that destroys minimality; without this, the extension from the manifold examples to Carnot groups is not yet load-bearing.

Authors: We appreciate this comment, which highlights an important point for rigor. Upon re-examination, we agree that the lifting argument would benefit from more explicit verification. In the revised manuscript, we will add a new subsection in the lifting section that provides a detailed proof. Specifically, we demonstrate that minimality is preserved because any competing horizontal curve in the Carnot group would project down to a horizontal curve in the original manifold with the same or shorter length, contradicting the minimality of the original geodesic. The strict abnormality is preserved due to the properties of the lift, which maintains the same control system structure. We will also verify that the endpoint map remains consistent under the lifting construction. This addresses the concern directly. revision: yes

Circularity Check

0 steps flagged

Explicit constructions and standard lifting yield self-contained results with no circular reduction

full rationale

The paper's central claims rest on explicit constructions of real-analytic sub-Riemannian manifolds that admit strictly abnormal minimizing geodesics exhibiting interior regularity loss and branching. These properties are verified directly from the given structures and the lifting procedure to Carnot groups, which is presented as a standard technique preserving horizontality, length, and abnormality without introducing fitted parameters or self-referential definitions. No equations reduce by construction to their inputs, no uniqueness theorems are imported from overlapping prior work as load-bearing, and no ansatzes are smuggled via self-citation. The derivation chain is therefore independent of the target claims and self-contained against the provided examples.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard axioms of real-analytic manifolds, sub-Riemannian distributions, and the lifting construction from simpler spaces; no free parameters or new entities are introduced.

axioms (1)

standard math Standard axioms and definitions of real-analytic manifolds and sub-Riemannian structures
The paper invokes the usual category of real-analytic sub-Riemannian manifolds without additional ad-hoc assumptions stated in the abstract.

pith-pipeline@v0.9.0 · 5362 in / 1138 out tokens · 49364 ms · 2026-05-15T00:48:22.302844+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Carnot groups admitting minimizing geodesics that are non-smooth at an interior point ... and that exhibit branching

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

[1]

Agrachev, D

A. Agrachev, D. Barilari, and U. Boscain.A comprehensive introduction to sub-Riemannian geometry, volume 181 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2020. With an appendix by Igor Zelenko

work page 2020
[2]

A. A. Agrachev and Y. L. Sachkov.Control theory from the geometric viewpoint, volume 87 ofEncyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, 2004. Control Theory and Optimization, II

work page 2004
[3]

A. A. Agrachev and A. V. Sarychev. Abnormal sub-Riemannian geodesics: Morse index and rigidity.Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire, 13(6):635–690, 1996

work page 1996
[4]

Barilari, Y

D. Barilari, Y. Chitour, F. Jean, D. Prandi, and M. Sigalotti. On the regularity of abnormal minimizers for rank 2 sub-Riemannian structures.J. Math. Pures Appl. (9), 133:118–138, 2020

work page 2020
[5]

Bianchini and F

S. Bianchini and F. Cavalletti. The Monge problem for distance cost in geodesic spaces.Comm. Math. Phys., 318(3):615–673, 2013

work page 2013
[6]

Boarotto, R

F. Boarotto, R. Monti, and A. Socionovo. Higher order Goh conditions for singular extremals of corank 1.Arch. Rational Mech. Anal., 248(23), 2024

work page 2024
[7]

Borza, M

S. Borza, M. Magnabosco, T. Rossi, and K. Tashiro. Measure contraction property and curvature-dimension condition on sub-Finsler Heisenberg groups.arXiv preprint 2402.14779, 2024

work page arXiv 2024
[8]

Cavalletti and M

F. Cavalletti and M. Huesmann. Existence and uniqueness of optimal transport maps.Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 32(6):1367–1377, 2015. INTERIOR SINGULARITY AND BRANCHING OF GEODESICS 37

work page 2015
[9]

Chitour, F

Y. Chitour, F. Jean, R. Monti, L. Rifford, L. Sacchelli, M. Sigalotti, and A. Socionovo. Not all sub-Riemannian minimizing geodesics are smooth.arXiv preprint 2501.18920, 2025

work page arXiv 2025
[10]

A. B. da Silva, A. Figalli, A. Parusi´ nski, and L. Rifford. Strong Sard conjecture and regularity of singular minimizing geodesics for analytic sub-Riemannian structures in dimension 3.Invent. Math., 229(1):395–448, 2022

work page 2022
[11]

N. Gigli. Optimal maps in non branching spaces with Ricci curvature bounded from below.Geom. Funct. Anal., 22(4):990–999, 2012

work page 2012
[12]

B. S. Goh. Necessary conditions for singular extremals involving multiple control variables.SIAM J. Control, 4:716–731, 1966

work page 1966
[13]

Hakavuori and E

E. Hakavuori and E. Le Donne. Non-minimality of corners in subriemannian geometry.Invent. Math., 206(3):693– 704, 2016

work page 2016
[14]

Hakavuori and E

E. Hakavuori and E. Le Donne. Blowups and blowdowns of geodesics in Carnot groups.J. Differential Geom., 123(2):267–310, 2023

work page 2023
[15]

Jean.Control of nonholonomic systems: from sub-Riemannian geometry to motion planning

F. Jean.Control of nonholonomic systems: from sub-Riemannian geometry to motion planning. SpringerBriefs in Mathematics. Springer, Cham, 2014

work page 2014
[16]

Le Donne.Metric Lie groups—Carnot-Carath´ eodory spaces from the homogeneous viewpoint, volume 306 of Graduate Texts in Mathematics

E. Le Donne.Metric Lie groups—Carnot-Carath´ eodory spaces from the homogeneous viewpoint, volume 306 of Graduate Texts in Mathematics. Springer, Cham, 2025

work page 2025
[17]

Le Donne, S

E. Le Donne, S. Nicolussi Golo, and N. Paddeu. Normal curves in sub-Finsler Lie groups: Branching for strongly convex norms and face stability for polyhedral norms.arXiv preprint 2510.26261, 2025

work page arXiv 2025
[18]

Le Donne, N

E. Le Donne, N. Paddeu, and A. Socionovo. Metabelian distributions andC 1 regularity of sub-Riemannian geodesics.arXiv preprint 2405.14997, 2024

work page arXiv 2024
[19]

G. P. Leonardi and R. Monti. End-point equations and regularity of sub-Riemannian geodesics.Geom. Funct. Anal., 18(2):552–582, 2008

work page 2008
[20]

Liu and H

W. Liu and H. J. Sussman. Shortest paths for sub-Riemannian metrics on rank-two distributions.Mem. Amer. Math. Soc., 118(564):x+104, 1995

work page 1995
[21]

Magnabosco and T

M. Magnabosco and T. Rossi. Failure of the curvature-dimension condition in sub-Finsler manifolds.arXiv preprint 2307.01820, 2023

work page arXiv 2023
[22]

Mietton and L

T. Mietton and L. Rizzi. Branching geodesics in sub-Riemannian geometry.Geom. Funct. Anal., 30(4):1139–1151, 2020

work page 2020
[23]

Montgomery

R. Montgomery. Abnormal minimizers.SIAM J. Control Optim., 32(6):1605–1620, 1994

work page 1994
[24]

Montgomery.A tour of subriemannian geometries, their geodesics and applications, volume 91 ofMathematical Surveys and Monographs

R. Montgomery.A tour of subriemannian geometries, their geodesics and applications, volume 91 ofMathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2002

work page 2002
[25]

Monti, A

R. Monti, A. Pigati, and D. Vittone. Existence of tangent lines to Carnot-Carath´ eodory geodesics.Calc. Var. Partial Differential Equations, 57(3):Paper No. 75, 18, 2018

work page 2018
[26]

T. Rad´ o. The isoperimetric inequality and the Lebesgue definition of surface area.Trans. Amer. Math. Soc., 61:530–555, 1947

work page 1947
[27]

Rifford.Sub-Riemannian geometry and optimal transport

L. Rifford.Sub-Riemannian geometry and optimal transport. SpringerBriefs in Mathematics. Springer, Cham, 2014

work page 2014
[28]

Socionovo

A. Socionovo. Sharp regularity for sub-Riemannian geodesics.Preprint arXiV, 2025

work page 2025
[29]

R. S. Strichartz. Sub-Riemannian geometry.J. Differential Geom., 24(2):221–263, 1986

work page 1986
[30]

R. S. Strichartz. Corrections to: Sub-Riemannian geometry.J. Differential Geom., 30(2):595–596, 1989. Universit`a degli Studi dell’Aquila, Via Vetoio, 67100 L’Aquila AQ, IT Email address:tommaso.rossi1@univaq.it Scuola Internazionale Superiore di Studi Avanzati (SISSA), Via Bonomea, 265, 34136 Trieste TS, IT Email address:aschiavo@sissa.it Unit´e de Math´...

work page 1989