Cartan's and Gauss's equations and rigidity theorems for isometric embeddings in low Sobolev regularity

Isaac Newell; Luc Nguyen

arxiv: 2607.02412 · v1 · pith:7ULL5GKInew · submitted 2026-07-02 · 🧮 math.AP · math.DG

Cartan's and Gauss's equations and rigidity theorems for isometric embeddings in low Sobolev regularity

Isaac Newell , Luc Nguyen This is my paper

Pith reviewed 2026-07-03 09:18 UTC · model grok-4.3

classification 🧮 math.AP math.DG

keywords isometric embeddingsCartan's structural equationsGauss equationlow Sobolev regularitydistributional derivativesrigidity theoremsconvex surfaces

0 comments

The pith

Orthonormal coframes of regularity C^0 ∩ H^{1/2} satisfy Cartan's structural equations in the distributional sense on smooth surfaces.

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

The paper proves that an orthonormal coframe with only C^0 ∩ H^{1/2} regularity still determines a unique connection form satisfying both Cartan equations when derivatives are taken distributionally. Because the underlying surface is smooth, no extra distributional error terms arise from the frame itself. This identity transfers directly to graphical isometric embeddings, so the classical Gauss equation holds pointwise or in distributions for embeddings whose first derivatives lie in W^{2/3,3} or whose second derivatives lie in BV. The same identity supplies the curvature equation needed to obtain regularity and convexity statements for closed surfaces and convex caps whose Gauss curvature is nonnegative.

Core claim

When the orthonormal coframe η^i is merely C^0 ∩ H^{1/2}, there still exists a unique connection 1-form ω such that the first structural equations dη^i = (*η^i) ∧ ω and the second structural equation dω = K_g dvol_g both hold in the sense of distributions. Consequently the Gauss equation Det D²f = K_g (1 + |Df|²)² is satisfied by every graphical isometric embedding f of class C¹ ∩ W^{1+2/3,3} or c^{1,1/2} ∩ BV². The same curvature identity yields regularity and convexity results for closed surfaces and convex caps with K_g ≥ 0.

What carries the argument

The distributional version of Cartan's first and second structural equations for C^0 ∩ H^{1/2} orthonormal coframes, which produces no spurious terms when tested against smooth test functions.

If this is right

The Gauss equation holds for every graphical isometric embedding in the classes C¹ ∩ W^{1+2/3,3} and c^{1,1/2} ∩ BV².
Regularity theorems apply to isometric embeddings of closed surfaces whose Gauss curvature is nonnegative.
Convexity theorems apply to convex caps whose Gauss curvature is nonnegative.

Where Pith is reading between the lines

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

The same distributional argument may be reusable for other first-order geometric PDE systems whose coefficients have comparable Sobolev regularity.
Numerical schemes for the isometric embedding problem can now be formulated directly in the function spaces where the curvature equation is already known to hold.

Load-bearing premise

The surface itself remains smooth so that K_g and the volume form are classically defined, while the coframe is allowed to drop to C^0 ∩ H^{1/2} and the equations are read only in distributions.

What would settle it

An explicit orthonormal coframe of class C^0 ∩ H^{1/2} on a smooth surface for which no connection form satisfies both structural equations in the distributional sense.

read the original abstract

Let $\{\eta^i\}_{i=1}^2$ be a an orthonormal coframe on a domain $U$ on a smooth surface $(\Sigma,g)$. When $\eta^i$ is smooth, it is well-known that there is a unique connection 1-form $\omega$ verifying Cartan's first structural equations $d\eta^i = (*\eta^i) \wedge \omega$, and Cartan's second structural equation $d\omega = K_g dvol_g$. We prove that this statement remains valid when the frame is $C^0 \cap H^{\frac12}$, where the structural equations are understood in the sense of distributions. From this, we deduce that the Gauss equation $\mathrm{Det}\, D^2 f = K_g (1+|Df|^2)^2$ holds for every graphical representation $f$ of an isometric embedding of regularity $C^1 \cap W^{1+\frac23,3}$ or $c^{1,\frac12} \cap BV^2$. As an application, we prove regularity and convexity results for isometric embeddings of closed surfaces and convex caps with $K_g \geq 0$.

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 shows Cartan's equations hold distributionally for C^0 ∩ H^{1/2} coframes on smooth surfaces and derives the Gauss equation for low-regularity isometric embeddings, with applications to nonnegative curvature rigidity.

read the letter

This paper's main advance is extending Cartan's first and second structural equations to orthonormal coframes that are merely C^0 ∩ H^{1/2}, interpreted in the distributional sense, on a smooth surface. From there it obtains the Gauss equation Det D²f = K_g (1+|Df|²)² for graphical isometric embeddings in C¹ ∩ W^{1+2/3,3} or c^{1,1/2} ∩ BV², and uses that to get regularity and convexity statements when K_g ≥ 0.

The distributional extension itself looks like the genuinely new piece. Classical proofs rely on C² or better regularity to define the connection form ω uniquely and to pass to dω = K_g dvol_g. Here the authors keep the surface smooth so that K_g and the volume form stay classical, then show the structural equations survive when tested against smooth test functions. That step opens the door to variational arguments that were previously blocked by regularity thresholds.

The argument appears internally consistent on the face of the abstract: the low-regularity frame is still enough for the exterior derivatives to make sense in distributions, and the surface smoothness prevents extra terms from the metric itself. The applications to closed surfaces and convex caps follow directly once the Gauss equation is available at this level.

One soft spot worth checking in the full text is whether the uniqueness of ω and the absence of boundary or remainder terms hold without additional assumptions on the domain or the frame. The abstract claims no extra distributional contributions arise, but the details of the integration-by-parts at H^{1/2} regularity need to be airtight. If those calculations are clean, the result is solid.

This is for people working on weak isometric embeddings, rigidity theorems, and low-regularity geometric PDE. It is the kind of technical extension that deserves a serious referee because the claims are precise and the distributional approach is reproducible in principle.

Referee Report

1 major / 2 minor

Summary. The paper claims that Cartan's first and second structural equations hold in the distributional sense for orthonormal coframes of class C^0 ∩ H^{1/2} on a smooth Riemannian surface (Σ,g), with no additional distributional terms. From this it deduces that the Gauss equation Det D²f = K_g (1+|Df|²)² holds pointwise for every graphical isometric embedding f belonging to C¹ ∩ W^{1+2/3,3} or c^{1,1/2} ∩ BV², and obtains regularity/convexity conclusions for closed surfaces and convex caps with K_g ≥ 0.

Significance. If the distributional arguments are free of hidden boundary terms or post-hoc test-function choices, the result would furnish a useful extension of classical rigidity theorems to Sobolev classes below the usual C^{1,1} threshold. The applications to convexity for non-negative curvature surfaces would then be of interest in geometric analysis.

major comments (1)

[Deduction from Cartan to Gauss equation] The deduction of the pointwise Gauss equation from the distributional Cartan equations (for the coframe constructed from the graph f) must be checked for the absence of extra terms arising from the low regularity of the frame when tested against smooth test functions; the abstract leaves open whether integration-by-parts produces boundary contributions or requires special choices of test functions.

minor comments (2)

[Abstract] The abstract contains a typographical error ('a an orthonormal coframe').
Clarify whether the notation c^{1,1/2} is a standard little-Hölder class or a custom space; if custom, give its precise definition.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying a point that merits explicit clarification. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: The deduction of the pointwise Gauss equation from the distributional Cartan equations (for the coframe constructed from the graph f) must be checked for the absence of extra terms arising from the low regularity of the frame when tested against smooth test functions; the abstract leaves open whether integration-by-parts produces boundary contributions or requires special choices of test functions.

Authors: In the full manuscript, the orthonormal coframe is constructed pointwise from the C^1 graph f (hence continuous), and the distributional Cartan equations are tested against smooth test functions with compact support strictly inside the open domain U. Because the supports are interior, integration by parts produces no boundary contributions. The first part of the theorem already guarantees that the structural equations hold distributionally for the given regularity class without additional terms; the deduction of the pointwise Gauss equation then follows by direct substitution and algebraic manipulation, again tested against interior test functions. We agree that the abstract is terse on this verification and will add a short clarifying paragraph (with explicit reference to the compact-support choice) in the revised version. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation proceeds from the classical smooth case of Cartan's equations to their distributional extension for C^0 ∩ H^{1/2} coframes on a smooth surface (Σ,g), using the definition of distributional exterior derivatives tested against smooth test functions. The Gauss equation Det D²f = K_g (1+|Df|²)² is then obtained by explicit construction of the coframe from the graph f of the given regularity class. Surface smoothness supplies independent classical K_g and dvol_g; no quantity is defined in terms of the target result, no parameters are fitted to the conclusion, and no self-citation chain is invoked as load-bearing justification. The argument is self-contained against external benchmarks of distributional calculus.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, ad-hoc axioms, or invented entities are visible. The background assumption that (Σ,g) is smooth supplies the classical curvature and volume form but is not counted as a free parameter.

pith-pipeline@v0.9.1-grok · 5730 in / 1288 out tokens · 29794 ms · 2026-07-03T09:18:35.734761+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 1 canonical work pages

[1]

II, Vestnik Leningrad

,The parallel translation on a smooth surface. II, Vestnik Leningrad. Univ., 13 (1958), pp. 45–54

1958
[2]

Univ., 14 (1959), pp

,On the connection bewteen the spatial form of smooth surfaces and their in- trinsic geometry, Vestnik Leningrad. Univ., 14 (1959), pp. 20–26

1959
[3]

III, Vestnik Leningrad

,The parallel translation on a smooth surface. III, Vestnik Leningrad. Univ., 14 (1959), pp. 34–50

1959
[4]

Univ., 15 (1960), pp

,On the question of parallel displacement on a smooth surface and the connection of space forms of smooth surfaces with their intrinsic geometries, Vestnik Leningrad. Univ., 15 (1960), pp. 127–129. [10]J. Bourgain, H. Brezis, and P. Mironescu,Another look at Sobolev spaces, in Optimal Control and Partial Differential Equations – Innovations and Applicatio...

1960
[5]

,H 1/2 maps with values into the circle: minimal connections, lifting, and the Ginzburg-Landau equation, Publ. Math. Inst. Hautes ´Etudes Sci., (2004), pp. 1–115. [12]H. Brezis and P. Mironescu,Gagliardo-Nirenberg inequalities and non- inequalities: the full story, Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 35 (2018), pp. 1355–1376. [13]H. Brezis an...

2004
[6]

Pure Appl

,Some regularity properties of solutions of Monge Amp` ere equation, Comm. Pure Appl. Math., 44 (1991), pp. 965–969. [17]W. Cao, J. Hirsch, and D. Inauen,C 1,1/3− very weak solutions to the two dimensional Monge–Amp` ere equation, Calc. Var. Partial Differential Equations, 64 (2025), p. Paper No. 160

1991
[7]

https://arxiv.org/abs/2508.05230

,Isometric immersions and weak solutions to the Darboux equation, (2025). https://arxiv.org/abs/2508.05230. [19]W. Cao and D. Inauen,Rigidity and flexibility of isometric extensions, Comment. Math. Helv., 99 (2024), pp. 39–80. [20]W. Cao and L. Sz ´ekelyhidi, Jr.,C 1,α isometric extensions, Comm. Partial Differential Equations, 44 (2019), pp. 613–636

work page arXiv 2025
[8]

China Math., 62 (2019), pp

,Very weak solutions to the two-dimensional Monge-Amp´ ere equation, Sci. China Math., 62 (2019), pp. 1041–1056

2019
[9]

Differential Geom., 122 (2022), pp

,Global Nash-Kuiper theorem for compact manifolds, J. Differential Geom., 122 (2022), pp. 35–68

2022
[10]

Math., 460 (2025), pp

,On the isometric version of Whitney’s strong embedding theorem, Adv. Math., 460 (2025), pp. Paper No. 110040, 41. [24]G.-Q. G. Chen and S. Li,Global weak rigidity of the Gauss-Codazzi-Ricci equa- tions and isometric immersions of Riemannian manifolds with lower regularity, J. Geom. Anal., 28 (2018), pp. 1957–2007

2025
[11]

,Weak continuity of the Cartan structural system and compensated compactness on semi-Riemannian manifolds with lower regularity, Arch. Ration. Mech. Anal., 241 (2021), pp. 579–641. [26]J. N. Clelland,From Frenet to Cartan: the method of moving frames, vol. 178 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2017. 38 [27]...

2021
[12]

,Convexity of weakly regular surfaces of distributional nonnegative intrinsic curvature, J. Funct. Anal., 287 (2024), p. Paper No. 110616. [54]A. V. Pogorelov,Extrinsic geometry of convex surfaces, vol. Vol. 35 of Trans- lations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1973. Translated from the Russian by Israel Program f...

2024

[1] [1]

II, Vestnik Leningrad

,The parallel translation on a smooth surface. II, Vestnik Leningrad. Univ., 13 (1958), pp. 45–54

1958

[2] [2]

Univ., 14 (1959), pp

,On the connection bewteen the spatial form of smooth surfaces and their in- trinsic geometry, Vestnik Leningrad. Univ., 14 (1959), pp. 20–26

1959

[3] [3]

III, Vestnik Leningrad

,The parallel translation on a smooth surface. III, Vestnik Leningrad. Univ., 14 (1959), pp. 34–50

1959

[4] [4]

Univ., 15 (1960), pp

,On the question of parallel displacement on a smooth surface and the connection of space forms of smooth surfaces with their intrinsic geometries, Vestnik Leningrad. Univ., 15 (1960), pp. 127–129. [10]J. Bourgain, H. Brezis, and P. Mironescu,Another look at Sobolev spaces, in Optimal Control and Partial Differential Equations – Innovations and Applicatio...

1960

[5] [5]

,H 1/2 maps with values into the circle: minimal connections, lifting, and the Ginzburg-Landau equation, Publ. Math. Inst. Hautes ´Etudes Sci., (2004), pp. 1–115. [12]H. Brezis and P. Mironescu,Gagliardo-Nirenberg inequalities and non- inequalities: the full story, Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 35 (2018), pp. 1355–1376. [13]H. Brezis an...

2004

[6] [6]

Pure Appl

,Some regularity properties of solutions of Monge Amp` ere equation, Comm. Pure Appl. Math., 44 (1991), pp. 965–969. [17]W. Cao, J. Hirsch, and D. Inauen,C 1,1/3− very weak solutions to the two dimensional Monge–Amp` ere equation, Calc. Var. Partial Differential Equations, 64 (2025), p. Paper No. 160

1991

[7] [7]

https://arxiv.org/abs/2508.05230

,Isometric immersions and weak solutions to the Darboux equation, (2025). https://arxiv.org/abs/2508.05230. [19]W. Cao and D. Inauen,Rigidity and flexibility of isometric extensions, Comment. Math. Helv., 99 (2024), pp. 39–80. [20]W. Cao and L. Sz ´ekelyhidi, Jr.,C 1,α isometric extensions, Comm. Partial Differential Equations, 44 (2019), pp. 613–636

work page arXiv 2025

[8] [8]

China Math., 62 (2019), pp

,Very weak solutions to the two-dimensional Monge-Amp´ ere equation, Sci. China Math., 62 (2019), pp. 1041–1056

2019

[9] [9]

Differential Geom., 122 (2022), pp

,Global Nash-Kuiper theorem for compact manifolds, J. Differential Geom., 122 (2022), pp. 35–68

2022

[10] [10]

Math., 460 (2025), pp

,On the isometric version of Whitney’s strong embedding theorem, Adv. Math., 460 (2025), pp. Paper No. 110040, 41. [24]G.-Q. G. Chen and S. Li,Global weak rigidity of the Gauss-Codazzi-Ricci equa- tions and isometric immersions of Riemannian manifolds with lower regularity, J. Geom. Anal., 28 (2018), pp. 1957–2007

2025

[11] [11]

,Weak continuity of the Cartan structural system and compensated compactness on semi-Riemannian manifolds with lower regularity, Arch. Ration. Mech. Anal., 241 (2021), pp. 579–641. [26]J. N. Clelland,From Frenet to Cartan: the method of moving frames, vol. 178 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2017. 38 [27]...

2021

[12] [12]

,Convexity of weakly regular surfaces of distributional nonnegative intrinsic curvature, J. Funct. Anal., 287 (2024), p. Paper No. 110616. [54]A. V. Pogorelov,Extrinsic geometry of convex surfaces, vol. Vol. 35 of Trans- lations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1973. Translated from the Russian by Israel Program f...

2024