{"paper":{"title":"Diffeomorphic Logarithm of Special Orthogonal Matrices","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Kyle A. Gallivan, P.-A. Absil, Wen Huang, Zhifeng Deng","submitted_at":"2026-06-17T16:00:29Z","abstract_excerpt":"The special orthogonal group $\\mathbb{SO}_n$ is a Lie group whose geometry and local structure are encoded by the exponential map in its Lie algebra $\\mathbf{Skew}_n$, the set of skew-symmetric matrices. The associated multi-valued inverse problem -- the matrix logarithm -- in $\\mathbb{SO}_n$ exhibits a highly nontrivial local diffeomorphism structure, which differs from the matrix logarithm for invertible matrices. This work characterizes the local diffeomorphism structure of the exponential in the set of skew-symmetric matrices where its derivative is invertible. We show that this set with a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.19223","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.19223/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}