{"paper":{"title":"Fr\\\"olicher-Nijenhuis cohomology on $G_2$- and ${\\rm Spin}(7)$-manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"H\\^ong V\\^an L\\^e, Kotaro Kawai, Lorenz Schwachh\\\"ofer","submitted_at":"2017-03-15T13:05:10Z","abstract_excerpt":"In this paper we show that a parallel differential form $\\Psi$ of even degree on a Riemannian manifold allows to define a natural differential both on $\\Omega^\\ast(M)$ and $\\Omega^\\ast(M, TM)$, defined via the Fr\\\"olicher-Nijenhuis bracket. For instance, on a K\\\"ahler manifold, these operators are the complex differential and the Dolbeault differential, respectively. We investigate this construction when taking the differential w.r.t. the canonical parallel $4$-form on a $G_2$- and ${\\rm Spin}(7)$-manifold, respectively. We calculate the cohomology groups of $\\Omega^\\ast(M)$ and give a partial"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.05133","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}