1 · The essentials
A surface’s rules for measuring lengths can be specified without first placing it in ordinary three-dimensional space. The manuscript constructs smooth, positive rules of this kind on an open square. At the square’s center, these rules agree with those of a flat plane to every order of differentiation. Nevertheless, no region around that center, however small, can be smoothly placed into three-dimensional space while preserving those length rules. This gives a negative answer to the question of whether every smooth surface metric can be realized locally in three-dimensional space. The main construction and its obstruction to such a realization have a Lean formalization.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine a sheet supplied with instructions telling you how to measure lengths at every point. The result gives instructions that match a flat sheet to every order at the center, yet cannot be followed by any smooth placement of even a small surrounding patch in three-dimensional space.
Why it matters
The result separates having smooth, positive length measurements from being able to realize them as a surface in three-dimensional space. Shrinking the region does not remove the obstruction. Even agreement with the flat metric to every order at one point does not guarantee a local realization.
2 · Going further
The local isometric realization problem asks whether a given surface metric can be realized by a smooth isometric immersion into Euclidean three-space. Here the domain is , and the manuscript constructs a smooth positive-definite Riemannian metric for which no neighborhood of the origin admits such an immersion. An isometric immersion must preserve the lengths specified by .
The construction has an additional feature: agrees with the Euclidean metric to every order at the origin. Thus its values and derivatives of every order there match those of the flat metric. This agreement at the origin coexists with failure of smooth isometric immersion on every neighborhood of that point.
The source describes this as a negative answer to the unrestricted smooth local isometric realization problem for surfaces. The abstract identifies the construction and its properties, but does not describe the mechanism of the obstruction or the steps of the argument. The main result is Lean-formalized; only this main manuscript is supplied, so no relationship to companion manuscripts can be described.
Useful notions
- Smooth functions and derivatives
- Length measurements on a surface
- Euclidean geometry
- Neighborhoods of a point
- Maps that preserve lengths
What was known before / what is new
Before
The source frames the question as the unrestricted smooth local isometric realization problem for surfaces. It supplies no earlier results or historical account, so no more specific prior state of knowledge is asserted here.
New
The manuscript constructs a smooth positive-definite metric on with no smooth isometric immersion into Euclidean three-space on any neighborhood of the origin. The metric nevertheless agrees with the Euclidean metric to every order at that point. The main result has a Lean formalization.
Limits and assumptions
The statement concerns one constructed smooth positive-definite metric, smooth isometric immersions, and the target space . Its obstruction holds on every neighborhood of the origin, not merely on the whole square. The supplied source identifies the main result as Lean-formalized but gives no finer breakdown of the formalization. It does not state conclusions for less regular immersions, other target dimensions, or additional hypotheses on metrics, and it lists no remaining open questions.
Mini glossary
- Riemannian metric —
- A smoothly varying rule for measuring lengths and angles in the tangent directions of a surface.
- Positive-definite —
- A metric is positive-definite when every nonzero tangent direction has strictly positive squared length.
- Smooth isometric immersion —
- A smooth map with injective derivative at each point that preserves the length measurements specified by the metric.
- Neighborhood of the origin —
- A region containing an open patch around the point .
- Agreement to every order —
- Two smooth metrics agree to every order at a point when their values and derivatives of every order coincide there.
Keywords : Riemannian metric · Isometric immersion · Local realization
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Constructs a smooth positive-definite metric on for which no neighborhood of the origin admits a smooth isometric immersion into ℝ3. This answers the unrestricted smooth local isometric realization problem for surfaces negatively, even after shrinking the neighborhood.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
A Smooth Metric with No Local Isometric Immersion into Three-Space
24/09/2026We construct a smooth positive-definite Riemannian metric on that agrees with the Euclidean metric to every order at the origin, yet no neighborhood of the origin admits a smooth isometric immersion into Euclidean three-space. This gives a negative answer to the unrestricted smooth local isometric realization problem for surfaces.