1 · The essentials
A space of vectors comes with a rule for measuring lengths and distances. Its unit sphere consists of all vectors whose length is one. Tingley’s problem asks whether a distance-preserving match between two such spheres must come from a match between their entire spaces. For nonzero complete vector spaces over the real numbers, the main result says yes, provided the sphere match reaches every point of the other sphere. The extension preserves distances, addition, and multiplication by real numbers, and it reaches every point of the other space. There is exactly one such extension, with no restriction on dimension. The main result has a Lean formalization.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine matching every location on two boundary maps while keeping all pairwise distances unchanged. The result says that, for the spaces under consideration, such a boundary match uniquely fixes a compatible match of the whole spaces; the image is imperfect because these spheres need not resemble ordinary round boundaries.
Why it matters
The result connects distance information on a unit sphere with the linear structure of an entire Banach space. Under the stated hypotheses, knowing the sphere up to a distance-preserving, onto match determines the space up to linear isometry. The absence of a dimension restriction is part of the conclusion.
2 · Going further
Let and be nonzero real Banach spaces, with unit spheres and . The precise question concerns a surjective isometry : a map reaching all of and satisfying for all . The main result states that extends uniquely to a surjective real-linear isometry . There is no dimension restriction.
The source identifies this extension question as Tingley’s problem and presents the manuscript as an affirmative solution. The distinction between sphere distances and distances throughout the space is central: the initial map gives distance preservation only for vectors of length one, whereas the conclusion covers vectors of all lengths and requires compatibility with the real vector-space operations.
The abstract describes an argument by contradiction centered on distances between different radii. If those distances are not preserved, the argument realizes a positive maximal defect in a possibly enlarged pair of Banach spaces. It then aligns extremal chords using common supports and Darbo’s fixed-point theorem. Support and convexity estimates yield the contradiction. These are the argument ingredients reported by the abstract; the supplied text does not give their detailed construction.
Only one manuscript is supplied, so no relationship to companion manuscripts can be described from this source. The main result has a Lean formalization. The supplied information does not specify how the individual ingredients of the abstract’s argument are represented within that formalization.
Useful notions
- Vectors and real-linear maps
- Norms and distances
- Completeness of a normed space
- Unit spheres
- Surjective maps and isometries
What was known before / what is new
Before
The source identifies Tingley’s problem as the question of extending a surjective sphere isometry to a surjective real-linear isometry of the whole spaces. It supplies no account of earlier partial results or historical chronology.
New
The main manuscript gives an affirmative answer for nonzero real Banach spaces without a dimension restriction. The extension is both surjective and unique. The main result has a Lean formalization.
Limits and assumptions
The stated result assumes nonzero real Banach spaces and a distance-preserving sphere map that is onto. It does not state a conclusion for maps that fail these hypotheses, nor does it state a complex-linear extension result. Lean formalization is reported for the main result, but the supplied source does not delineate coverage of every auxiliary claim or argument step. No remaining open question is specified in the supplied text.
Mini glossary
- Real Banach space —
- A real vector space equipped with a norm in which every Cauchy sequence converges.
- Unit sphere —
- The set of vectors whose norm is one.
- Surjective isometry —
- A map that preserves distances and reaches every point of its target.
- Real-linear isometry —
- A distance-preserving map that respects vector addition and multiplication by real numbers.
- Extension —
- A map on a larger domain that agrees with the original map wherever the original map is defined.
Keywords : Tingley’s problem · Banach spaces · Sphere isometries
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)
Resolves Tingley's problem: every surjective isometry between the unit spheres of nonzero real Banach spaces extends uniquely to a surjective real-linear isometry of the whole spaces. No dimension restriction is imposed, so the metric geometry of the unit sphere determines the Banach space up to linear isometry.
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 positive solution to Tingley’s problem
23/09/2026Every surjective isometry between the unit spheres of real Banach spaces extends uniquely to a surjective real-linear isometry, giving an affirmative solution to Tingley's problem. If distances between different radii are not preserved, we realize the positive maximal defect in a possibly enlarged pair of Banach spaces. We then align extremal chords using common supports and Darbo's fixed-point theorem and obtain a contradiction from support and convexity estimates.