Pith Number

pith:ID6FSNHH

pith:2026:ID6FSNHHX6TTJNAOANDTUBVMJN

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An isometric immersion of a flat Klein bottle into Euclidean 3-space

Stepan Paul

A flat Klein bottle admits an explicit piecewise-linear isometric immersion into three-dimensional Euclidean space.

arxiv:2605.16730 v1 · 2026-05-16 · math.MG

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4 Citations open

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Claims

C1strongest claim

We present an explicit piecewise linear map from a flat Klein bottle into Euclidean 3-space that is an isometric immersion -- a path isometry that is locally injective. The image is a self-intersecting polyhedron with embedded vertex figures where each vertex has zero angle defect.

C2weakest assumption

The construction of the map enforces the path isometry property so long as certain numerically-verifiable inequalities are satisfied, and we show that checking the local injectivity property at each vertex via another set of inequalities suffices.

C3one line summary

An explicit piecewise-linear isometric immersion of the flat Klein bottle into R^3 is constructed as a self-intersecting polyhedron with zero angle defect at vertices.

References

14 extracted · 14 resolved · 0 Pith anchors

[1] Flat tori in three-dimensional space and convex integration.Proceedings of the National Academy of Sciences, 109(19):7218–7223, 2012 2012

[2] Ulrich Brehm. Oberwolfach Report, 1978 1978

[3] American Mathematical Society Providence, 2001 2001

[4] Polyhedral realizations of developments.Vestnik Leningrad 1960

[5] Isometric piecewise-linear embeddings of two-dimensional manifolds with a polyhe- dral metric intoR 3.Algebra i Analiz, 7(3):76–95, 1995 1995

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:02:38.802965Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

40fc5934e7bfa734b40e03473a06ac4b7a7d9a5eb9aadd07842faa36937ceed5

Aliases

arxiv: 2605.16730 · arxiv_version: 2605.16730v1 · doi: 10.48550/arxiv.2605.16730 · pith_short_12: ID6FSNHHX6TT · pith_short_16: ID6FSNHHX6TTJNAO · pith_short_8: ID6FSNHH

Agent API

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/ID6FSNHHX6TTJNAOANDTUBVMJN \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 40fc5934e7bfa734b40e03473a06ac4b7a7d9a5eb9aadd07842faa36937ceed5

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "cce1c70651052217186f671a8e4976fc14afcf455d953f0b6fcbafe9071907a1",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.MG",
    "submitted_at": "2026-05-16T00:46:47Z",
    "title_canon_sha256": "b9e5a6bd876585f476ebc2d140dbf7c5534d74d2770a3e053e997e1fd0656a36"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.16730",
    "kind": "arxiv",
    "version": 1
  }
}