A smooth surface metric that cannot fit locally in three-space

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Family 334 · Differential geometry
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-26

1. The question

A small piece of a smooth surface seems like something that ought to fit into three-dimensional space. But suppose the starting object is an abstract rule for measuring lengths, rather than a surface already sitting in space. Must some sufficiently small neighborhood have a smooth realization with exactly those measurements?

Family 334 claims that the answer is no. It constructs a smooth positive-definite metric on a square for which no neighborhood of the origin admits a smooth isometric immersion into $\mathbb R^3$.

The proposed example has a further surprise: at the origin, the metric agrees with the ordinary Euclidean metric to every derivative order. No finite Taylor expansion there detects a difference, and even the full formal Taylor series is the same. The obstruction is encoded in increasingly small surrounding regions.

This is a local statement. It does not rely on holes, global topology, completeness, or the difficulty of closing a surface up. It concerns whether a single neighborhood can be realized smoothly at all.

2. Background and essential definitions

A Riemannian metric on a planar domain is a smooth positive-definite symmetric matrix field $g=(g_{ij})$. A tangent vector $v$ has squared length $v^{\mathsf T}gv$. Positive definiteness ensures that every nonzero vector has positive length. It says nothing about the sign of the metric’s Gaussian curvature.

A map $F=(F_1,F_2,F_3)$ is an isometric immersion if

$$\langle\partial_iF,\partial_jF\rangle=g_{ij} \qquad(i,j\in\{1,2\}).$$

The derivative must have rank two. An immersion is locally an embedding after shrinking around a point, so the local existence questions for smooth immersions and smooth embeddings have the same content here. This differs from global problems, where self-intersections matter.

The Gaussian curvature $K$ is determined by the metric and its derivatives. Positive curvature resembles a sphere locally, negative curvature a saddle, and zero curvature marks a potentially degenerate transition. These descriptions are geometric intuition, not complete local classifications.

Smooth means $C^\infty$. Analytic means that a function equals a convergent power series near every point. Analytic functions are smooth, but smooth functions can be invisible to their Taylor series at one point. For example,

$$\psi(r)=\begin{cases}e^{-1/r^2},&r\ne0,\\0,&r=0\end{cases}$$

has every derivative zero at zero but is positive elsewhere. This illustrates the kind of distinction that makes agreement of all Taylor coefficients compatible with different nearby geometry.

The full Taylor jet of a metric at a point is the collection of all derivatives of its coefficients there. Equality with the Euclidean jet means that the difference from $du^2+dv^2$ has every partial derivative zero at the origin. It does not mean that the metric equals the Euclidean metric on a neighborhood.

3. Development history and previous partial results

The analytic local problem was settled by the Janet–Cartan theorem: analytic surface metrics have local isometric realizations in three-space. Smooth metrics with nonzero Gaussian curvature also admit local smooth realizations through equations of elliptic or hyperbolic type. The difficult cases arise where curvature vanishes and the equation degenerates or changes type.

Lin established foundational results for nonnegative curvature and for clean changes of sign, with finite-differentiability estimates. A clean sign change means that $K=0$ but its differential is nonzero at the point. Nakamura and Maeda obtained smooth local existence under that condition. Later work treated more complicated zero sets, including transverse curves and cusps, under additional finite-order and geometric assumptions.

Regularity must be tracked on both sides of these results. A theorem producing a $C^m$ immersion for each finite $m$, on neighborhoods that may shrink with $m$, does not automatically supply one $C^\infty$ immersion on one fixed neighborhood. At lower regularity, Nash–Kuiper flexibility gives $C^1$ realizations, but these do not carry the second-order bending information used in a smooth obstruction.

Pogorelov constructed a $C^{2,1}$ metric with no local $C^2$ realization. The notation means that second derivatives are Lipschitz continuous. Nadirashvili and Yuan’s published refinement achieved such a counterexample with nonnegative curvature. These input metrics are less regular than the smooth metric claimed in M1.

There is a historical discrepancy that M1 explicitly leaves unresolved. A 2002 Nadirashvili–Yuan preprint states a smooth nonpositively curved counterexample with no local $C^3$ embedding. Their later published paper, and later literature cited by M1, describe the unrestricted smooth question as still open. The present digest reports those different statements without deciding between them. M1 does not use the earlier nonsolvability claim; it offers its own analytical obstruction.

4. What the new manuscript claims

M1 states that there is a smooth positive-definite metric $g$ on $(-1,1)^2$ for which no open neighborhood $U$ of the origin admits a smooth isometric immersion

$$F:(U,g)\longrightarrow\mathbb R^3.$$

The metric can additionally be chosen so that

$$\partial^\alpha(g_{ij}-\delta_{ij})(0)=0$$

for every multi-index $\alpha$ and every $i,j$. A multi-index records how many derivatives are taken in each coordinate, and $\delta_{ij}$ are the coefficients of the Euclidean metric.

The construction uses both positive and negative Gaussian curvature. It therefore does not settle the corresponding smooth local-existence question under either global sign restriction $K\geq0$ or $K\leq0$.

An intermediate statement obstructs certain scalar height functions on an entire fixed square. These heights must satisfy the Darboux equation, have gradient norm below one, have a nonzero selected diagonal Hessian entry, and have a sufficiently small mixed-to-diagonal Hessian ratio. The manuscript finds obstructing metrics densely, in the stronger sense of a residual set, within a specified space of perturbations.

This intermediate claim does not exclude every locally defined height on every tiny neighborhood of that square. In fact, its curvature zeros are clean sign changes, compatible with known pointwise local-existence theorems. The final construction is needed to ensure that any hypothetical immersion near the origin would induce one of the prohibited heights on a full selected square.

5. Why the advance matters

Importance: 5/5. The family claims a negative resolution of the unrestricted smooth local realization problem in three-space, a central boundary case in the geometry of isometric immersions.

Its force comes from combining maximum regularity of the metric with a genuinely local obstruction. There is no rough corner in the metric, no global topological obstruction, and no prescribed boundary behavior making the problem artificially restrictive.

Agreement with the entire Euclidean Taylor jet gives a particularly clear lesson about smooth geometry. Formal information at one point can fail to determine whether a neighborhood admits a smooth geometric realization. The arrangement of nearby regions matters even when every derivative measured exactly at the distinguished point looks flat.

The family complements the four-dimensional claim in family 333. An additional ambient direction changes the realization problem substantially. This comparison concerns the stated theorems; the digest does not independently establish their validity or settle the conflicting historical accounts.

6. The main idea

Reduce an immersion to a scalar equation

Choose a fixed unit vector $e$ in ambient three-space. Any hypothetical immersion gives a height function $z=\langle F,e\rangle$. Its covariant Hessian is the surface-corrected second derivative: in coordinates,

$$H_{ij}=\partial_i\partial_jz-\Gamma^r_{ij}\partial_rz,$$

where the Christoffel coefficients $\Gamma^r_{ij}$ are determined by the metric, and the repeated index $r$ is summed. These corrections make the Hessian independent of the chosen coordinate description.

The Gauss equation, relating intrinsic curvature to extrinsic bending, implies the necessary Darboux equation

$$H_{11}H_{22}-H_{12}^2 =K\det(g)\bigl(1-|dz|_g^2\bigr).$$

The source uses only necessity: excluding an appropriate height excludes the corresponding immersion. It does not need to reconstruct an immersion from every solution of this scalar equation.

Obstruct a height across a whole patch

The model patch has a negative-curvature center and a positive-curvature exterior. Two cap regions provide estimates on opposite sides of a thin central slab. M1 claims that fixed low-order control of an admissible height, together with all-order control of the metric, yields all-order interior control on the caps. No high-order boundary data are imposed on the artificial edges facing the central region.

The manuscript then perturbs the metric itself in a thin oscillating strip. This is important: constructing an arbitrary unsolvable second-order equation would not suffice, because the Darboux coefficients must come from one actual Riemannian metric.

An integral test isolates the high-frequency effect forced by the strip. The strip width is chosen proportional to a small parameter divided by its oscillation frequency. The desired forcing is linear in that parameter, while leading competing terms are quadratic; smooth endpoint data have a small high-frequency contribution. M1 uses this separation of scales to exclude heights in each fixed bounded class.

A Baire-category argument then removes the a priori bounds. In this setting, a residual set is a countable intersection of open dense sets. One can organize candidate heights into countably many bounded classes and avoid all of them simultaneously. This is an existence argument about a complete function-space topology, not a probability distribution over metrics.

Make every hypothetical immersion encounter an obstruction

The geometric assembly places smaller obstructed patches around the boundary of a negatively curved disk. A boundary-saddle argument forces any hypothetical graph realization to have a boundary point where its second fundamental form is nonzero and rank one.

Taking the height in the normal direction at that point gives zero first derivative and a Hessian equal to the second fundamental form. In coordinates aligned with its kernel, the mixed Hessian entry vanishes while a transverse diagonal entry remains nonzero. A nearby orientation from a fixed finite collection retains the strict inequalities required by the patch obstruction.

This is why the argument uses patches approaching boundary points in several orientations. The immersion gets to choose its bending directions, so the metric construction must ensure that at least one whole patch detects whichever direction occurs.

Finally, disks with their surrounding patches accumulate at the origin. Their amplitudes are chosen sufficiently small that the total metric remains positive definite and smooth, with the Euclidean Taylor jet at the origin. Any neighborhood of the origin contains one of these complete configurations. A hypothetical smooth immersion there would produce a forbidden height on one of its patches, giving the claimed contradiction.

7. Scope, limitations, and open questions

The asserted obstruction is to smooth immersions for a smooth metric. It should not be silently strengthened to nonexistence at a specified finite differentiability class. Conversely, it does not obstruct the low-regularity realizations furnished by $C^1$ flexibility.

The metric has mixed-sign curvature. Smooth local realization under either one-sided curvature restriction remains outside this theorem’s conclusion. Positive definiteness of the metric must not be confused with positive curvature.

The whole-square analytical obstruction and the final local obstruction have different roles. The former concerns a restricted class of heights on a fixed domain; the latter follows only after the geometric construction forces any immersion to produce such a height somewhere. Retaining that distinction prevents an apparent conflict with known local existence at clean curvature zeros.

The source does not claim uniform estimates as the curvature amplitude of its shrinking patches tends to zero. Nor does it offer a simple finite formula from which a practical obstruction test for arbitrary smooth metrics follows. Classification of realizable smooth metrics, finite-regularity thresholds, and sign-restricted problems remain separate directions.

8. Sources and further reading

• Ghomi, Open problems in geometry of curves and surfaces, for the local realization question.
• Nadirashvili and Yuan, Improving Pogorelov’s isometric embedding counterexample, for the published finite-regularity result and historical discussion.
• Nadirashvili and Yuan, Counterexamples for local isometric embedding, version 1, for the earlier claim whose status M1 leaves unresolved.
• Optional background: isometric immersions and the second fundamental form.

Source manuscripts

• M1. OpenAI, A Smooth Metric with No Local Isometric Immersion into Three-Space (September 24, 2026).

Sources consulted on 26 October 2026 at repository revision 1581d4c. Download the manuscript citations.