{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2004:MLERTPFYCJFVFXBYY2DRZGYWIL","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"ebaf61bb855882213999b42653a5cd88f976e148d6b8ce0ba3590cc2bc3f3189","cross_cats_sorted":["math.AP"],"license":"","primary_cat":"math.DG","submitted_at":"2004-09-17T15:16:55Z","title_canon_sha256":"27a289670360e3009e45cf20b3876e73af8dcd8281ecc84eaa5a5b09cd1cde83"},"schema_version":"1.0","source":{"id":"math/0409303","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0409303","created_at":"2026-05-18T01:05:25Z"},{"alias_kind":"arxiv_version","alias_value":"math/0409303v2","created_at":"2026-05-18T01:05:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0409303","created_at":"2026-05-18T01:05:25Z"},{"alias_kind":"pith_short_12","alias_value":"MLERTPFYCJFV","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_16","alias_value":"MLERTPFYCJFVFXBY","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_8","alias_value":"MLERTPFY","created_at":"2026-05-18T12:25:52Z"}],"graph_snapshots":[{"event_id":"sha256:9b01453ac1ffb73e5572c636f499a507e2b0d8f817eb1ca1770a5fa4347ff605","target":"graph","created_at":"2026-05-18T01:05:25Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"The $L^2$-metric or Fubini-Study metric on the non-linear Grassmannian of all submanifolds of type $M$ in a Riemannian manifold $(N,g)$ induces geodesic distance 0. We discuss another metric which involves the mean curvature and shows that its geodesic distance is a good topological metric. The vanishing phenomenon for the geodesic distance holds also for all diffeomorphism groups for the $L^2$-metric.","authors_text":"David Mumford, Peter W. Michor","cross_cats":["math.AP"],"headline":"","license":"","primary_cat":"math.DG","submitted_at":"2004-09-17T15:16:55Z","title":"Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0409303","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:4c0dc700048453cff33a6ab9e31cea441a71c1703d5116a961f1b14a5c890a50","target":"record","created_at":"2026-05-18T01:05:25Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"ebaf61bb855882213999b42653a5cd88f976e148d6b8ce0ba3590cc2bc3f3189","cross_cats_sorted":["math.AP"],"license":"","primary_cat":"math.DG","submitted_at":"2004-09-17T15:16:55Z","title_canon_sha256":"27a289670360e3009e45cf20b3876e73af8dcd8281ecc84eaa5a5b09cd1cde83"},"schema_version":"1.0","source":{"id":"math/0409303","kind":"arxiv","version":2}},"canonical_sha256":"62c919bcb8124b52dc38c6871c9b1642d238d973da465fce9941c48b86e1ac86","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"62c919bcb8124b52dc38c6871c9b1642d238d973da465fce9941c48b86e1ac86","first_computed_at":"2026-05-18T01:05:25.980734Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:25.980734Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Rzz7fc60XiGrltM9y5Oh8gRpkPEegKZ837bEKSGgsRCAT5tjxxH5w1HSc0JGt1A9uVvMYDXWy5sd/3owkFKkAA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:25.981362Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0409303","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4c0dc700048453cff33a6ab9e31cea441a71c1703d5116a961f1b14a5c890a50","sha256:9b01453ac1ffb73e5572c636f499a507e2b0d8f817eb1ca1770a5fa4347ff605"],"state_sha256":"b46edf0a9feed9dd09c6def7e625b5fcc1895825b6bac8a02161eb99ccfcad44"}