A smooth metric with no local immersion into three-space

Source and verification

This article explains family 334 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

🏷️ Preliminaries: intrinsic metrics and scalar heights

A smooth positive definite metric $g=(g_{ij})$ on a planar domain specifies lengths and angles without assuming the domain is already a surface in Euclidean three-space. A smooth isometric immersion would satisfy

$$⟨{\partial }_{i}F,{\partial }_{j}F⟩=g_{ij}.$$

Positive definiteness makes its differential injective. An immersion becomes an embedding after restricting to a sufficiently small neighborhood of each point, so the two local existence questions agree. Smooth maps are the subject here; the $C^1$ category has different flexibility.

Manuscript result

There is a smooth positive definite metric on $(-1,1{)}^2$ for which no neighborhood of the origin admits a smooth isometric immersion into ${\mathbb{R}}^3$. The metric can have exactly the same derivatives of every order at the origin as the Euclidean metric. (OpenAI, 2026)

The construction uses both signs of Gaussian curvature. Positive definiteness does not mean positive curvature, and the result does not settle the smooth problem under $K\ge 0$ or $K\le 0$.

The prerequisites are surface geometry, second-order PDE, smooth-function compactness, and Baire category. The proof first obstructs a scalar height on a whole square, then forces any hypothetical immersion near the origin to yield such a height.

🏷️ The necessary Darboux equation

Fix an ambient unit vector $e$ and let $z=⟨F,e⟩$. For a local unit normal $n$, put $\nu =⟨n,e⟩$. The Gauss formula and orthogonal decomposition of $e$ give

$${\nabla }_g^{2}z=\nu \mathrm{II},|dz{|}_g^2+{\nu }^2=1.$$

Here $\mathrm{II}$ is the scalar second fundamental form. The Gauss equation $\det \mathrm{II}=K\det g$ now yields the Darboux equation

$$\det ({\nabla }_g^{2}z)=K\det (g)(1-|dz{|}_g^2).$$

Write

$$\begin{array}{rl}{H}_{ij}& =z_{ij}-{\Gamma }_{ij}^r{z}_r,\\ E& =\det (g)(1-|dz{|}_g^2).\end{array}$$

Then the equation is $H_{xx}{H}_{yy}-H_{xy}^2=KE$. Only necessity is used; no converse reconstruction from a scalar solution is required.

If $e$ is normal at a point $b$, then $dz(b)=0$ and ${\nabla }_g^{2}z(b)=\mathrm{II}(b)$. This will connect the geometric and analytic arguments.

🏷️ Obstruction on a whole square

The model square is $S=[-3,3{]}^2$, with background curvature

$$\begin{array}{c}{K}_0(x,y)=\kappa (x^2-h(y)),\\ \kappa >0.\end{array}$$

The even function $h$ equals one near zero, is positive for $|y|<1$, and changes sign cleanly at $y=\pm 1$. Perturbations are supported in a small central box that remains negatively curved.

An admissible height is smooth on a neighborhood of the entire closed square and satisfies

$$\begin{array}{c}{H}_{xx}{H}_{yy}-H_{xy}^2=KE,\\ E>0,H_{yy}\ne 0,\\ |H_{xy}/H_{yy}|\le 1/100.\end{array}$$

The patch theorem says that metrics admitting no admissible height form a residual subset of the permitted smooth perturbations. Hence obstructed metrics occur in every prescribed smooth neighborhood of the background.

The whole-square requirement matters. Curvature zeros on these patches are clean sign changes. Local solutions around individual points need not extend to one height satisfying the inequalities on the whole square, so the patch obstruction does not contradict those local existence results.

🏷️ Caps and two-sided regularity

Fix a class of candidates with bounded $C^8$ norm and positive lower bounds for the metric, $E$, and $|H_{yy}|$. Put

$$q=H_{xy}/H_{yy},D={\partial }_x-q{\partial }_y.$$

Flow coordinates straighten $D$. The small bound on $q$ keeps these coordinates inside controlled margins.

The cap estimate upgrades the low-order bounds to bounds on every derivative in upper and lower cap regions whenever the metric has corresponding bounds there. An artificial cap edge may lie in negative curvature; no high-order boundary data are imposed on it.

Differentiating the equation produces a transverse drift involving $K_y$. A directed multiplier uses this drift and a small tangential term where $K_y$ vanishes. A weighted elliptic estimate controls the remaining positive-curvature region. These estimates supply smooth data on two cuts facing the central negative-curvature slab.

In the slab the equation is treated in a hyperbolic direction. Energy estimates propagate smooth tangential data toward both edges of a thin perturbation. Spatial intervals shrink faster than the characteristic speeds, controlling boundary flux. This two-sided information is what prevents an arbitrary candidate height from absorbing the perturbation through uncontrolled endpoint oscillations.

🏷️ An oscillating metric pulse

Suppose a bounded class of metrics with admissible heights were dense in a nonempty open set. Fix $g_{\ast }$ there and shear coordinates to $(\xi ,\theta )$ so that $\theta$ is a hyperbolic time direction.

Perturb the metric itself by

$${\tilde{g}}_{\tau }=g_{\ast }+A_{\tau }d{\theta }^2,$$

where, for large frequency $\tau$,

$$A_{\tau }={\tau }^{-N}\chi (\xi )\varphi (\tau \theta /\delta )\cos (\tau \xi ).$$

The cutoff $\varphi$ is nonnegative with positive integral. The pulse has width $\delta /\tau$. Choose the fixed integer $N$ larger than the finitely many derivative orders defining the selected open neighborhood. Then the test metrics belong to that neighborhood for large $\tau$; convergence in every smooth seminorm is not required.

Perturbing the metric keeps all Darboux coefficients geometrically compatible. The leading change in $K\det g$ is $-\frac{1}{2}{\partial }_{\xi }^2{A}_{\tau }$, of size ${\tau }^{2-N}$. A nonsolvable general Monge–Ampère equation with unrelated coefficients would not provide this metric obstruction.

Compare a candidate height with a finite Taylor construction for the unperturbed equation, matching the lower-edge Cauchy data. Their difference has zero initial data and a forced equation. Test it against $\chi (\xi )\cos (\tau \xi )$ and integrate through the pulse.

The forcing contributes an absolute moment at least $(c+o(1))\delta {\tau }^{1-N}$. The other spatial terms contribute at most $(C+o(1)){\delta }^2{\tau }^{1-N}$. The source size times the pulse width explains the first scale; the short-time energy estimate supplies the extra factor of $\delta$ in the competing terms. Smooth endpoint data have a negligible high-frequency moment by repeated integration by parts.

Choose $\delta$ small enough that $C\delta <c/2$, then increase $\tau$. The linear and quadratic contributions cannot balance. The forcing coefficient has one sign on each strip and is bounded away from zero; it need not converge or keep the same sign for different frequencies.

🏷️ Baire category removes the a priori bounds

Every admissible height on the compact square has some finite $C^8$ bound and positive lower bounds for its nonvanishing quantities. Countably many classes, also indexed by rational approximations to the central slope ratio, cover all such heights.

The pulse argument makes each class nowhere dense. The smooth metric-perturbation space is a Baire space, so the complement of their union is residual. This is an existence construction in a function space, not a closed formula for a single obstructed patch.

A hypothetical immersion near the origin need not initially provide the required nonzero Hessian entry and small mixed ratio. The next geometric argument forces them.

🏷️ Boundary saddles and accumulation

Place disjoint negatively curved disks approaching the origin, with zero curvature on their boundaries. Around every boundary point, arrange shrinking obstructed patches in each orientation of a finite angular net.

A graph immersion of a negatively curved disk cannot have its second fundamental form vanish on the entire boundary. Indeed, write the graph as $u$ over a planar domain. Boundary vanishing of $\mathrm{II}$ gives $D^{2}u=0$ there, so $Du$ is constant along the connected boundary. Subtracting an affine function gives $v=0$ on the boundary and

$$\det D^{2}v<0\text{in the interior}.$$

A nonzero $v$ has a positive interior maximum or negative interior minimum. Its Hessian is then semidefinite, with nonnegative determinant, a contradiction. Thus some boundary point has nonzero second fundamental form. Zero boundary curvature makes it rank one.

Use the fixed ambient normal at that point as the height direction. The height has zero differential and rank-one Hessian there. Align one coordinate axis with the Hessian kernel; a nearby orientation in the finite net preserves a nonzero transverse entry and a mixed-to-transverse ratio below $1/100$.

Continuity preserves the strict inequalities on a neighborhood. One full obstructed patch of that orientation lies there, so the height would be admissible on a patch chosen to admit none.

Every immersion near the origin becomes a graph after shrinking its domain. The disks accumulate at the origin, so one disk and its needed patches lie inside that domain. This yields the contradiction for every candidate neighborhood.

🏷️ Smoothness and the Euclidean Taylor jet

First realize a small prescribed curvature $K_0$ by

$$\bar{g}=d{X}^2+f(X,Y{)}^{2}d{Y}^2,$$

using the ODE

$$\begin{array}{c}{f}_{XX}=-K_{0}f,\\ f(0,Y)=1,f_X(0,Y)=0.\end{array}$$

Small curvature amplitude keeps $f$ uniformly positive, and direct calculation gives $K_{\bar{g}}=-f_{XX}/f$.

Patch corrections must be controlled after rescaling to the original coordinates. A patch of scale $\lambda$ can amplify an order-$m$ tensor derivative by ${\lambda }^{-m-2}$. The residual patch theorem allows arbitrarily small perturbations in each prescribed finite list of seminorms. The $n$th correction can therefore be made at most $2^{-n}$ in every order $m\le n$, after paying these scaling factors.

Every fixed derivative series then has a uniformly summable tail. Each summand vanishes on a neighborhood of the origin. The curvature and tensor corrections are consequently flat there, and the ODE propagates flatness to $f-1$. All derivatives of $g-\delta$ vanish at the origin despite the obstruction.

A smooth germ therefore contains information absent from its Taylor series. Likewise, solutions at each finite differentiability order on shrinking neighborhoods would not give one smooth realization on a fixed neighborhood.

🏷️ Key steps and proof outlines

Curvature differentiation supplies the missing transverse control

The cap regularity argument does not assume smooth boundary data on its artificial edge. Its central identity comes from the covariant Hessian. With $D={\partial }_x-q{\partial }_y$ and $q=H_{xy}/H_{yy}$, one has $H({\partial }_y,D)=0$ and $H(D,D)=KE/H_{yy}$. Commute one covariant derivative of these identities. In dimension two the commutator is proportional to Gaussian curvature, so the remainder is $O(K)$. Dividing by the nonvanishing $H_{yy}$ gives a transverse drift proportional to $K_y$.

More concretely, after solving the Darboux equation for $z_{xx}=P$ and differentiating $l$ times in the original $y$ coordinate, the straightened equation for $u={\partial }_y^{l}z$ has the form

$$u_{tt}+A{u}_{ss}+B{u}_t+C{u}_s=R_l,A=G_{1}K,C=(l+1)A_s+O_l(A),$$

with $G_1$ uniformly positive. The derivative is taken before changing coordinates: high derivatives of the flow chart are not yet controlled. This avoids using the regularity being proved as an input. (OpenAI, 2026)

A derivative count keeps the remainder below top order

In the differentiated nonlinear equation, a factor containing a derivative of $z$ of order at least $l-1$ consumes at least $l-3$ differentiations. For $l\ge 8$, two such factors would require more than $l$ differentiations. Thus at most one factor needs an $L^2$ estimate; all others have lower order and are bounded in supremum norm by local Sobolev embedding. This gives an $L^2$ bound for $R_l$ from the preceding induction level.

A directed multiplier exploits the $(l+1)A_s$ term through the sign-changing curvature region; the positive-curvature portion has an elliptic estimate. Together they gain one derivative on smaller cap rectangles. The margins of those rectangles are fixed independently of the candidate solution, so the induction supplies uniform bounds rather than merely individual smoothness.

The short pulse has one extra power of time in its response

In the central hyperbolic region, an equation of the form

$$v_{\theta \theta }-P_1{v}_{\theta \xi }-S_1{v}_{\xi \xi }=b^i{\partial }_{i}v+R$$

has energy density $(v_{\theta }^2+S_1{v}_{\xi }^2)/2$. Its flux is bounded by $(|P_1|+\sqrt{S_1})$ times that energy. Choose intervals shrinking faster than this speed; their moving boundaries then contribute a nonpositive term. Integration by parts and Gronwall control the square root of energy by its initial value plus $\int \Vert R{\Vert }_2$.

Apply this to the difference from the lower-edge Taylor construction. The pulse forcing has size ${\tau }^{2-N}$ and duration $\delta /\tau$, so its tested moment has scale $\delta {\tau }^{1-N}$. The induced competing response has an additional short-time factor $\delta$. Smooth data at both pulse edges have arbitrarily small high-frequency moments after repeated integration by parts. Choosing $\delta$ first and then $\tau$ makes the forced moment impossible to balance.

The Baire argument then removes the bounded-candidate hypotheses, and the accumulating-patch construction converts whole-square failure into failure of every neighborhood of the origin. These last two steps are logically separate: the analytic pulse alone does not already obstruct an arbitrary local immersion.

See Also

Smooth immersions into four-space treats the companion target dimension. Its additional normal direction supports a different correction mechanism.

🏷️ Manuscripts and verification scope

• A Smooth Metric with No Local Isometric Immersion into Three-Space — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The documented selection is the smooth local obstruction on every neighborhood of the origin. It does not explicitly list the additional Euclidean Taylor-jet strengthening. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026. A Smooth Metric with No Local Isometric Immersion into Three-Space.