{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:K2UE7DWYCJMJEJQTBXW2JT2N5V","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":"3feebd1f18dbb89037a79c324ea507426e64c1fcfd2257a14faca85ee12e9696","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2021-06-28T14:55:24Z","title_canon_sha256":"7e8a72dbce55684acbdef5957f15b22765dd8f0afe67965c376896d8764fda9f"},"schema_version":"1.0","source":{"id":"2106.14781","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2106.14781","created_at":"2026-05-20T01:06:02Z"},{"alias_kind":"arxiv_version","alias_value":"2106.14781v2","created_at":"2026-05-20T01:06:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2106.14781","created_at":"2026-05-20T01:06:02Z"},{"alias_kind":"pith_short_12","alias_value":"K2UE7DWYCJMJ","created_at":"2026-05-20T01:06:02Z"},{"alias_kind":"pith_short_16","alias_value":"K2UE7DWYCJMJEJQT","created_at":"2026-05-20T01:06:02Z"},{"alias_kind":"pith_short_8","alias_value":"K2UE7DWY","created_at":"2026-05-20T01:06:02Z"}],"graph_snapshots":[{"event_id":"sha256:e4c674ea7c3fd6682d09831431b8d05d17f7abaea5d24c7fe69f9f84d20d83da","target":"graph","created_at":"2026-05-20T01:06:02Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2106.14781/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this note, we derive explicit formulae for the curvature of a convex sum of Riemannian metrics, \\(g_t = (1-t)g_0 + t g_1\\). We study whether such a deformation can increase the \\emph{average} of the Riemann curvature component \\(R_t(X,Y,Y,X)\\) along an immersed, totally geodesic flat torus. Because a first-order increase is prohibited, we obtain necessary and sufficient conditions for \\(g_t\\) to have a positive average variation of order \\(r \\geq 2\\). These conditions are applied to paths joining \\(g_0\\) to classical metric deformations, including conformal changes, vertical warpings, and C","authors_text":"Giovane Galindo, Leonardo F. Cavenaghi, Llohann D. Speran\\c{c}a","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2021-06-28T14:55:24Z","title":"The curvature of convex sum of metrics and applications"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2106.14781","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":"sha256:6a6ace02dc483522098e1d6ccffb0f07eb97e783a0aec61682fb5555bdd6b8d2","target":"record","created_at":"2026-05-20T01:06:02Z","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":"3feebd1f18dbb89037a79c324ea507426e64c1fcfd2257a14faca85ee12e9696","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2021-06-28T14:55:24Z","title_canon_sha256":"7e8a72dbce55684acbdef5957f15b22765dd8f0afe67965c376896d8764fda9f"},"schema_version":"1.0","source":{"id":"2106.14781","kind":"arxiv","version":2}},"canonical_sha256":"56a84f8ed812589226130deda4cf4ded585254b0ec234566b9edfc2b976aba1f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"56a84f8ed812589226130deda4cf4ded585254b0ec234566b9edfc2b976aba1f","first_computed_at":"2026-05-20T01:06:02.594336Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T01:06:02.594336Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"v1R19lhhQAEM1rjwIj8Dx6utuHM7QK2sYe6B0KC28tPC6d1nJ0jAU4EkhlN8exnp9py+4elh24L96UE9Id+1Cg==","signature_status":"signed_v1","signed_at":"2026-05-20T01:06:02.595229Z","signed_message":"canonical_sha256_bytes"},"source_id":"2106.14781","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6a6ace02dc483522098e1d6ccffb0f07eb97e783a0aec61682fb5555bdd6b8d2","sha256:e4c674ea7c3fd6682d09831431b8d05d17f7abaea5d24c7fe69f9f84d20d83da"],"state_sha256":"6df2e7470a70a8539fe22e17f6458a08984a5317368197e83d69813f5647bb5c"}