{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:RFJTB4FTYPTMIBSCLYCMDIJRR3","short_pith_number":"pith:RFJTB4FT","canonical_record":{"source":{"id":"1206.4403","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-06-20T07:43:25Z","cross_cats_sorted":[],"title_canon_sha256":"0ced264f0877b44d3d3fa07314369e60d3ad6bac4d2a502c7cfdcd824bb58aba","abstract_canon_sha256":"c9690c51de43560eaa8a21a7dbc8060f55af8482aa74254338c745111a73ad4b"},"schema_version":"1.0"},"canonical_sha256":"895330f0b3c3e6c406425e04c1a1318ed1369b159e07f9acbf12b979f9087bd3","source":{"kind":"arxiv","id":"1206.4403","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.4403","created_at":"2026-05-18T03:53:05Z"},{"alias_kind":"arxiv_version","alias_value":"1206.4403v1","created_at":"2026-05-18T03:53:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.4403","created_at":"2026-05-18T03:53:05Z"},{"alias_kind":"pith_short_12","alias_value":"RFJTB4FTYPTM","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_16","alias_value":"RFJTB4FTYPTMIBSC","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_8","alias_value":"RFJTB4FT","created_at":"2026-05-18T12:27:20Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:RFJTB4FTYPTMIBSCLYCMDIJRR3","target":"record","payload":{"canonical_record":{"source":{"id":"1206.4403","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-06-20T07:43:25Z","cross_cats_sorted":[],"title_canon_sha256":"0ced264f0877b44d3d3fa07314369e60d3ad6bac4d2a502c7cfdcd824bb58aba","abstract_canon_sha256":"c9690c51de43560eaa8a21a7dbc8060f55af8482aa74254338c745111a73ad4b"},"schema_version":"1.0"},"canonical_sha256":"895330f0b3c3e6c406425e04c1a1318ed1369b159e07f9acbf12b979f9087bd3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:53:05.905338Z","signature_b64":"gqwyaksT9Jg1F54Gi9QW2aeeWdhoSZHziQEvP7c4lTeVfbURSMNu1h3HDPT+jWOkqUIYKuOY7oY9xD8Txku+Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"895330f0b3c3e6c406425e04c1a1318ed1369b159e07f9acbf12b979f9087bd3","last_reissued_at":"2026-05-18T03:53:05.904487Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:53:05.904487Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1206.4403","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:53:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4UBCJrJk5mV3y4v4v0lAtDv6wEI/j5TUIhsG84XswOhS5sMYivDj+8nzC06jrQtbfTIgQu0q0TijxH/S8n//BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T17:52:56.281459Z"},"content_sha256":"fc21da1eb36748cb7b1004c4466cd0d6807d02d070313051d39c2c85a94b607a","schema_version":"1.0","event_id":"sha256:fc21da1eb36748cb7b1004c4466cd0d6807d02d070313051d39c2c85a94b607a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:RFJTB4FTYPTMIBSCLYCMDIJRR3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Some Rigidity Conditions on Berwald Structures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Ricardo Gallego Torrom\\'e","submitted_at":"2012-06-20T07:43:25Z","abstract_excerpt":"This thesis contains an introduction to the method of average in Finsler geometry. The method is applied to Berwald spaces, obtaining geodesic rigidity conditions. We prove that the Levi-Civita connection of any Riemannian metric affine equivalent to the Berwald metric leaves invariant the indicatrix of th Finsler metric F. A converse result also holds: if (M,F) is a Finsler structure such that there is a Riemannian connection whose Levi-Civita leaves invariant by parallel transport the indicatrix of the Finsler structure, then the structure (M,F) is Berwald. As an application we obtain a nece"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.4403","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":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:53:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uYWwlTe4ieciBbM6YEWegyKNGUTGB7Xfeg3NKVa8UHSR6YFfcarGxR1+NWUF0sy4tqSlTIFSPrDtjdUu7LVSCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T17:52:56.281801Z"},"content_sha256":"a6ecfd493e38b03589ac7db12c0d4be5448462f4fcc5ed8844ea42f3414de94a","schema_version":"1.0","event_id":"sha256:a6ecfd493e38b03589ac7db12c0d4be5448462f4fcc5ed8844ea42f3414de94a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RFJTB4FTYPTMIBSCLYCMDIJRR3/bundle.json","state_url":"https://pith.science/pith/RFJTB4FTYPTMIBSCLYCMDIJRR3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RFJTB4FTYPTMIBSCLYCMDIJRR3/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-03T17:52:56Z","links":{"resolver":"https://pith.science/pith/RFJTB4FTYPTMIBSCLYCMDIJRR3","bundle":"https://pith.science/pith/RFJTB4FTYPTMIBSCLYCMDIJRR3/bundle.json","state":"https://pith.science/pith/RFJTB4FTYPTMIBSCLYCMDIJRR3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RFJTB4FTYPTMIBSCLYCMDIJRR3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:RFJTB4FTYPTMIBSCLYCMDIJRR3","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":"c9690c51de43560eaa8a21a7dbc8060f55af8482aa74254338c745111a73ad4b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-06-20T07:43:25Z","title_canon_sha256":"0ced264f0877b44d3d3fa07314369e60d3ad6bac4d2a502c7cfdcd824bb58aba"},"schema_version":"1.0","source":{"id":"1206.4403","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.4403","created_at":"2026-05-18T03:53:05Z"},{"alias_kind":"arxiv_version","alias_value":"1206.4403v1","created_at":"2026-05-18T03:53:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.4403","created_at":"2026-05-18T03:53:05Z"},{"alias_kind":"pith_short_12","alias_value":"RFJTB4FTYPTM","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_16","alias_value":"RFJTB4FTYPTMIBSC","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_8","alias_value":"RFJTB4FT","created_at":"2026-05-18T12:27:20Z"}],"graph_snapshots":[{"event_id":"sha256:a6ecfd493e38b03589ac7db12c0d4be5448462f4fcc5ed8844ea42f3414de94a","target":"graph","created_at":"2026-05-18T03:53:05Z","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":"This thesis contains an introduction to the method of average in Finsler geometry. The method is applied to Berwald spaces, obtaining geodesic rigidity conditions. We prove that the Levi-Civita connection of any Riemannian metric affine equivalent to the Berwald metric leaves invariant the indicatrix of th Finsler metric F. A converse result also holds: if (M,F) is a Finsler structure such that there is a Riemannian connection whose Levi-Civita leaves invariant by parallel transport the indicatrix of the Finsler structure, then the structure (M,F) is Berwald. As an application we obtain a nece","authors_text":"Ricardo Gallego Torrom\\'e","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-06-20T07:43:25Z","title":"Some Rigidity Conditions on Berwald Structures"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.4403","kind":"arxiv","version":1},"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:fc21da1eb36748cb7b1004c4466cd0d6807d02d070313051d39c2c85a94b607a","target":"record","created_at":"2026-05-18T03:53:05Z","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":"c9690c51de43560eaa8a21a7dbc8060f55af8482aa74254338c745111a73ad4b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-06-20T07:43:25Z","title_canon_sha256":"0ced264f0877b44d3d3fa07314369e60d3ad6bac4d2a502c7cfdcd824bb58aba"},"schema_version":"1.0","source":{"id":"1206.4403","kind":"arxiv","version":1}},"canonical_sha256":"895330f0b3c3e6c406425e04c1a1318ed1369b159e07f9acbf12b979f9087bd3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"895330f0b3c3e6c406425e04c1a1318ed1369b159e07f9acbf12b979f9087bd3","first_computed_at":"2026-05-18T03:53:05.904487Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:53:05.904487Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gqwyaksT9Jg1F54Gi9QW2aeeWdhoSZHziQEvP7c4lTeVfbURSMNu1h3HDPT+jWOkqUIYKuOY7oY9xD8Txku+Dw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:53:05.905338Z","signed_message":"canonical_sha256_bytes"},"source_id":"1206.4403","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fc21da1eb36748cb7b1004c4466cd0d6807d02d070313051d39c2c85a94b607a","sha256:a6ecfd493e38b03589ac7db12c0d4be5448462f4fcc5ed8844ea42f3414de94a"],"state_sha256":"bd7f8ca25397446e39b4e0d64ffaf4de6a67a9b7db5cde8461f5caf2e0667b76"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Tfcbyi74AfqDKch6s6hj06ttAfsxe9H/+kswWeAWZLtjSDIxMiekPaeETYHqPXT4hi+WX3I4ekSqnjDx7Cn7Bw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T17:52:56.283657Z","bundle_sha256":"9ca6cdea57fefbe14d868694b81b5be23c504c3d4dc6d511f604597857608448"}}