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The geodesic equation is the nonlinear, nonlocal wave equation $\\eta_{tt} = \\partial_s(\\sigma \\eta_s)$, with $\\lvert \\eta_s\\rvert\\equiv 1$ and $\\sigma$ given by $\\sigma_{ss}- \\lvert \\eta_{ss}\\rvert^2 \\sigma = -\\lvert \\eta_{st}\\rvert^2$, with boundary conditions $\\sigma(t,1)=\\sigma(t,-1)=0$ and $\\eta(t,0)=0$. We prove that the space of arcs is a submanifold of the space of all curves, "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1105.1754","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-05-09T18:50:17Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"2c46cc75d7155c167feca39f0395afc400b118c2427dcb96526764fb71649d39","abstract_canon_sha256":"b08c2391455d9e452f0052f7d9f6d23d5bd98d26a2aed8a19e15ba31d1147a29"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:22:32.562639Z","signature_b64":"8Jz+y/vhSJeX5CN0/Xm6QMnMH7RMTkgcgWbX6GZBwEcJp++47tYuVLpQHCKa4pOxgKk2d8wAm0qb/xNFEGsGCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c7a3dbba248c6acdac5b9834c05a25d30060b0e0c0f8fa2d49f7e9d9aa856b8b","last_reissued_at":"2026-05-18T04:22:32.562014Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:22:32.562014Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The geometry of whips","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Stephen C. Preston","submitted_at":"2011-05-09T18:50:17Z","abstract_excerpt":"In this paper we study geometric aspects of the space of arcs parametrized by unit speed in the $L^2$ metric. Physically this corresponds to the motion of a whip, and it also arises in studying shape recognition. The geodesic equation is the nonlinear, nonlocal wave equation $\\eta_{tt} = \\partial_s(\\sigma \\eta_s)$, with $\\lvert \\eta_s\\rvert\\equiv 1$ and $\\sigma$ given by $\\sigma_{ss}- \\lvert \\eta_{ss}\\rvert^2 \\sigma = -\\lvert \\eta_{st}\\rvert^2$, with boundary conditions $\\sigma(t,1)=\\sigma(t,-1)=0$ and $\\eta(t,0)=0$. 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