{"paper":{"title":"On the triviality of the generalized tangent bundle","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The generalized tangent bundle TM ⊕ T*M is trivial whenever the manifold is parallelizable, but the converse is false.","cross_cats":[],"primary_cat":"math.DG","authors_text":"Fernando Etayo, Pablo G\\'omez-Nicol\\'as, Rafael Santamar\\'ia","submitted_at":"2026-05-15T10:18:16Z","abstract_excerpt":"We study the relations between the triviality of the tangent bundle $TM$ and the generalized tangent bundle $\\mathbb{T}M = TM\\oplus T^*M$ of a manifold. We show that the generalized tangent bundle of a paralellizable manifold is trivial. We also prove that the converse implication does not hold, by studying the cases of the M\\\"obius strip, spheres and projective spaces. Finally, we relate the triviality of the generalized tangent bundle to generalized geometric structures."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"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.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"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.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The generalized tangent bundle TM ⊕ T*M is trivial whenever the manifold is parallelizable, but the converse is false.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"8e198ddfacc84f63921a93cb972196c3ac67e5e74d17100893790386bc7f92f0"},"source":{"id":"2605.15818","kind":"arxiv","version":1},"verdict":{"id":"1c214631-803a-4384-a9f5-b02ac56ba564","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T19:10:30.302340Z","strongest_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.","one_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.","pipeline_version":"pith-pipeline@v0.9.0","weakest_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.","pith_extraction_headline":"The generalized tangent bundle TM ⊕ T*M is trivial whenever the manifold is parallelizable, but the converse is false."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.15818/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T19:31:19.095107Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T19:21:20.112379Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T17:33:48.726419Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T17:21:55.877253Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"125f22c8ca761de0a45eb318eeff6e2295c87a6bb8eeec68d426bc3369171c3e"},"references":{"count":15,"sample":[{"doi":"","year":1980,"title":"J. Allard. (1980). Sums of stably trivial vector bundles.Math. Proc. Cambridge Philos. Soc.87(1), 97–107","work_id":"dce1c6ea-73d1-43ee-ab7f-b53d19eb8765","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2004,"title":"New aspects of the ddc-lemma","work_id":"b9c2a2e5-59cc-4155-9a7a-96d1c70f1361","ref_index":2,"cited_arxiv_id":"math/0501406","is_internal_anchor":true},{"doi":"","year":2006,"title":"G. Cavalcanti and M. Gualtieri. (2006). A surgery for generalized complex structures on 4-manifolds. J. Differential Geom.76(1), 35–43","work_id":"17f45d71-2d94-4692-a0cb-2bb01bb8c373","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1990,"title":"T. J. Courant. (1990). Dirac manifolds.Trans. Amer. Math. Soc.319(2), 631–661","work_id":"ee2c14e9-0e31-42bb-a41c-3d0cbd06c023","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"F. Etayo, P. G´ omez-Nicol´ as and R. Santamar´ ıa. (2024). 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