{"paper":{"title":"A vector field induced de Rham-Hodge theory on manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Any vector field on a manifold induces its own de Rham-Hodge theory by defining a modified inner product on forms.","cross_cats":[],"primary_cat":"math.DG","authors_text":"Zhe Su","submitted_at":"2026-05-15T05:48:32Z","abstract_excerpt":"We introduce a de Rham-Hodge framework induced by a vector field on a compact, oriented smooth manifold. By utilizing a vector field induced isomorphism on differential forms, we define a vector field induced Hodge $L^2$-inner product, codifferential, and Hodge Laplacian on differential forms. We then establish the resulting de Rham-Hodge theory for closed manifolds and extend it to manifolds with boundary by imposing certain vector field induced boundary conditions. We also include some remarks on this resulting framework."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We then establish the resulting de Rham-Hodge theory for closed manifolds and extend it to manifolds with boundary by imposing certain vector field induced boundary conditions.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"A given vector field on the manifold induces an isomorphism on the space of differential forms that can be used to define a new L2 inner product and the associated operators.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A vector field on a compact oriented manifold is used to induce an isomorphism on forms, from which a new L2 inner product, codifferential, and Hodge Laplacian are defined, yielding a de Rham-Hodge theory on closed manifolds and with boundary conditions on manifolds with boundary.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Any vector field on a manifold induces its own de Rham-Hodge theory by defining a modified inner product on forms.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"7953a66626c0c4147f1b84472bcbaf72ec7d84be37dfa0ff6a474f1c968d222f"},"source":{"id":"2605.15643","kind":"arxiv","version":1},"verdict":{"id":"9b27d70f-d8a0-41f4-8c7b-3665af080e9a","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T19:47:18.056683Z","strongest_claim":"We then establish the resulting de Rham-Hodge theory for closed manifolds and extend it to manifolds with boundary by imposing certain vector field induced boundary conditions.","one_line_summary":"A vector field on a compact oriented manifold is used to induce an isomorphism on forms, from which a new L2 inner product, codifferential, and Hodge Laplacian are defined, yielding a de Rham-Hodge theory on closed manifolds and with boundary conditions on manifolds with boundary.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"A given vector field on the manifold induces an isomorphism on the space of differential forms that can be used to define a new L2 inner product and the associated operators.","pith_extraction_headline":"Any vector field on a manifold induces its own de Rham-Hodge theory by defining a modified inner product on forms."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.15643/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T20:01:19.259061Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T20:01:16.824530Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T19:33:35.669656Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T17:21:56.095013Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"b5407a9a91f96843e377baab1b71e11d6402b78fa4a12eab81457ce57c7f4bf6"},"references":{"count":23,"sample":[{"doi":"","year":2012,"title":"Differential Geometry and its Applications30(2), 179–194 (2012)","work_id":"a9af8825-c583-4928-858c-58a9b429ed53","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1988,"title":"Publications Math´ ematiques de l’IH´ES68, 175–186 (1988)","work_id":"8824577e-cb1f-4c74-b13a-975464e38e8c","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2003,"title":"ACM Transactions on Graphics (TOG)22(1), 4–32 (2003)","work_id":"c1c5fb2a-8931-415a-b61e-5ce1a96ea8ae","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1983,"title":"In: S´ eminaire de Probabilit´ es XIX 1983/84: Proceedings, pp","work_id":"63a79aac-6c2b-4808-9d79-8f2b761dc3d7","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"Results in Mathematics79(5), 187 (2024)","work_id":"911a91ce-e86b-4dba-968f-28c455795268","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":23,"snapshot_sha256":"1f60106ac7aa5158c2b93eb73388c063e39c457386c7c3072bb2ce9b7a6cb6c6","internal_anchors":1},"formal_canon":{"evidence_count":2,"snapshot_sha256":"9ee391b2802438e638eceb97d535f7e5470ff6a64c6e6507552128a539dc6ecc"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}