Pith Number

pith:5MLW2OQL

pith:2026:5MLW2OQLZB2WNJCLZOB3RFTN6I

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On the triviality of the generalized tangent bundle

Fernando Etayo, Pablo G\'omez-Nicol\'as, Rafael Santamar\'ia

The generalized tangent bundle TM ⊕ T*M is trivial whenever the manifold is parallelizable, but the converse is false.

arxiv:2605.15818 v1 · 2026-05-15 · math.DG

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\pithnumber{5MLW2OQLZB2WNJCLZOB3RFTN6I}

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Record completeness

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3 Author claim open · sign in to claim

4 Citations open

5 Replications open

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The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

We show that the generalized tangent bundle of a parallelizable manifold is trivial. We also prove that the converse implication does not hold, by studying the cases of the Möbius strip, spheres and projective spaces.

C2weakest assumption

The triviality of the generalized tangent bundle can be verified independently of the tangent bundle triviality using the standard definition TM ⊕ T*M on the listed example manifolds without additional hidden assumptions on the manifold structure.

C3one line summary

The generalized tangent bundle of a parallelizable manifold is trivial, but the converse does not hold, as shown by the Möbius strip, spheres, and projective spaces; it is also related to generalized geometric structures.

References

15 extracted · 15 resolved · 2 Pith anchors

[1] J. Allard. (1980). Sums of stably trivial vector bundles.Math. Proc. Cambridge Philos. Soc.87(1), 97–107 1980

[2] New aspects of the ddc-lemma 2004 · arXiv:math/0501406

[3] G. Cavalcanti and M. Gualtieri. (2006). A surgery for generalized complex structures on 4-manifolds. J. Differential Geom.76(1), 35–43 2006

[4] T. J. Courant. (1990). Dirac manifolds.Trans. Amer. Math. Soc.319(2), 631–661 1990

[5] F. Etayo, P. G´ omez-Nicol´ as and R. Santamar´ ıa. (2024). Metric polynomial structures on generalized geometry.Publ. Math. Debrecen105(1-2), 171–196 2024

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:01:20.162080Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

eb176d3a0bc87566a44bcb83b8966df236b0a81c7fc3bdfbbc6c92b6b7a024d9

Aliases

arxiv: 2605.15818 · arxiv_version: 2605.15818v1 · doi: 10.48550/arxiv.2605.15818 · pith_short_12: 5MLW2OQLZB2W · pith_short_16: 5MLW2OQLZB2WNJCL · pith_short_8: 5MLW2OQL

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/5MLW2OQLZB2WNJCLZOB3RFTN6I \
  | 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: eb176d3a0bc87566a44bcb83b8966df236b0a81c7fc3bdfbbc6c92b6b7a024d9

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "104786246c633603db96eb6ca19da95434280166c5c0195986bff88064c09ddf",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.DG",
    "submitted_at": "2026-05-15T10:18:16Z",
    "title_canon_sha256": "dd95b2a4667a3ff0aff65c9bf5da52e27c059ec76c8cd9229b0c797e19885734"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.15818",
    "kind": "arxiv",
    "version": 1
  }
}