{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:VBPLUCAETPB43UPY2MFJWMGUBV","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":"1ed1b654784a549488de441a2b17a9afa421f70976be8b86a9ef73c54d6733e0","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2010-09-19T07:51:52Z","title_canon_sha256":"2428b3b0f82f9f392f9b8ef6f29d1001a3a9ca52f7e9f09d32edf41f9431cc0b"},"schema_version":"1.0","source":{"id":"1009.3616","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.3616","created_at":"2026-05-18T04:00:01Z"},{"alias_kind":"arxiv_version","alias_value":"1009.3616v3","created_at":"2026-05-18T04:00:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.3616","created_at":"2026-05-18T04:00:01Z"},{"alias_kind":"pith_short_12","alias_value":"VBPLUCAETPB4","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_16","alias_value":"VBPLUCAETPB43UPY","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_8","alias_value":"VBPLUCAE","created_at":"2026-05-18T12:26:15Z"}],"graph_snapshots":[{"event_id":"sha256:d2b5de0b3c9b59a50adecc83ab4802f2d3d1b6a2d966ab2024468cd3ac9a1ffe","target":"graph","created_at":"2026-05-18T04:00:01Z","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":"Let $M$ and $N$ be connected manifolds without boundary with $\\dim(M) < \\dim(N)$, and let $M$ compact. Then shape space in this work is either the manifold of submanifolds of $N$ that are diffeomorphic to $M$, or the orbifold of unparametrized immersions of $M$ in $N$. We investigate the Sobolev Riemannian metrics on shape space: These are induced by metrics of the following form on the space of immersions: $$ G^P_f(h,k) = \\int_{M} \\g(P^f h, k)\\, \\vol(f^*\\g)$$ where $\\g$ is some fixed metric on $N$, $f^*\\g$ is the induced metric on $M$, $h,k \\in \\Gamma(f^*TN)$ are tangent vectors at $f$ to the","authors_text":"Martin Bauer, Peter W. Michor, Philipp Harms","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2010-09-19T07:51:52Z","title":"Sobolev metrics on shape space of surfaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.3616","kind":"arxiv","version":3},"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:d9c614c897cbbf9c3ef8bd6f13ee6aa4ff10a098ab8380d3f43b6b4d3d5a2843","target":"record","created_at":"2026-05-18T04:00:01Z","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":"1ed1b654784a549488de441a2b17a9afa421f70976be8b86a9ef73c54d6733e0","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2010-09-19T07:51:52Z","title_canon_sha256":"2428b3b0f82f9f392f9b8ef6f29d1001a3a9ca52f7e9f09d32edf41f9431cc0b"},"schema_version":"1.0","source":{"id":"1009.3616","kind":"arxiv","version":3}},"canonical_sha256":"a85eba08049bc3cdd1f8d30a9b30d40d558875a0e073137b811084a8a489cf29","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a85eba08049bc3cdd1f8d30a9b30d40d558875a0e073137b811084a8a489cf29","first_computed_at":"2026-05-18T04:00:01.375769Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:00:01.375769Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"FMZ8hLNta4tcEpS1pnbRzdtRyyLLTT5ayRVW9opJWeG0qoZk/nHHOQR3Jjyb0RbHMe3K8k4J0z3WEEcOFGDgDw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:00:01.376263Z","signed_message":"canonical_sha256_bytes"},"source_id":"1009.3616","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d9c614c897cbbf9c3ef8bd6f13ee6aa4ff10a098ab8380d3f43b6b4d3d5a2843","sha256:d2b5de0b3c9b59a50adecc83ab4802f2d3d1b6a2d966ab2024468cd3ac9a1ffe"],"state_sha256":"824ac32acd9f1c537f98c20ba6c160b324f224dc9c73985d2933aefc2c71e01e"}