{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:OE7KDZYJUCVG4HJQXNNCQ4KWQG","short_pith_number":"pith:OE7KDZYJ","canonical_record":{"source":{"id":"1703.00341","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-03-01T15:20:49Z","cross_cats_sorted":[],"title_canon_sha256":"331d3c65207821e89fce3ffd0fbb451421b3858a4c02866a0dbec7a820837009","abstract_canon_sha256":"4a0c0778aede8ec7f6c92bd9eb526ee6ba9e5083beff88cfea94afc5ee0547da"},"schema_version":"1.0"},"canonical_sha256":"713ea1e709a0aa6e1d30bb5a2871568191a9fa15d90ea87eee51501f68610fed","source":{"kind":"arxiv","id":"1703.00341","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1703.00341","created_at":"2026-05-18T00:23:09Z"},{"alias_kind":"arxiv_version","alias_value":"1703.00341v3","created_at":"2026-05-18T00:23:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.00341","created_at":"2026-05-18T00:23:09Z"},{"alias_kind":"pith_short_12","alias_value":"OE7KDZYJUCVG","created_at":"2026-05-18T12:31:34Z"},{"alias_kind":"pith_short_16","alias_value":"OE7KDZYJUCVG4HJQ","created_at":"2026-05-18T12:31:34Z"},{"alias_kind":"pith_short_8","alias_value":"OE7KDZYJ","created_at":"2026-05-18T12:31:34Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:OE7KDZYJUCVG4HJQXNNCQ4KWQG","target":"record","payload":{"canonical_record":{"source":{"id":"1703.00341","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-03-01T15:20:49Z","cross_cats_sorted":[],"title_canon_sha256":"331d3c65207821e89fce3ffd0fbb451421b3858a4c02866a0dbec7a820837009","abstract_canon_sha256":"4a0c0778aede8ec7f6c92bd9eb526ee6ba9e5083beff88cfea94afc5ee0547da"},"schema_version":"1.0"},"canonical_sha256":"713ea1e709a0aa6e1d30bb5a2871568191a9fa15d90ea87eee51501f68610fed","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:23:09.140542Z","signature_b64":"LEBWhSxMBVDmiBOoxfqdNdRJwjWK+lxjXG5FamOuio9E7i58yAJ8hYuSY0/lLvvd+afFuhuLUXO/WCM9Zb9dDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"713ea1e709a0aa6e1d30bb5a2871568191a9fa15d90ea87eee51501f68610fed","last_reissued_at":"2026-05-18T00:23:09.139851Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:23:09.139851Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1703.00341","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:23:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CNnYwuGYv08giiyULKdtpSQazs67anZSJvgN+jTmyez/BvRhdbl/o9ACAL46YtT9hnez9vyniQqauyA+LOVIBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T14:26:52.686915Z"},"content_sha256":"cce8b5b0e068e8945146a9c5d1bf289f61f62b1b66a00ae80b8ebaa5cf6947ef","schema_version":"1.0","event_id":"sha256:cce8b5b0e068e8945146a9c5d1bf289f61f62b1b66a00ae80b8ebaa5cf6947ef"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:OE7KDZYJUCVG4HJQXNNCQ4KWQG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Geometry of Asymptotically harmonic manifolds with minimal horospheres","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Hemangi Shah","submitted_at":"2017-03-01T15:20:49Z","abstract_excerpt":"$(M^n,g)$ be a complete Riemannian manifold without conjugate points. In this paper, we show that if $M$ is also simply connected, then $M$ is flat, provided that $M$ is also asymptotically harmonic manifold with minimal horospheres (AHM). The (first order) flatness of $M$ is shown by using the strongest criterion: $\\{{e_i}\\}$ be an orthonormal basis of $T_{p}M$ and $\\{b_{e_{i}}\\}$ be the corresponding Busemann functions on $M$. Then, (1) The vector space $V = span\\{b_{v} | v \\in T_{p}M \\}$ is finite dimensional and dim $V = $ dim $M = n$.(2) $\\{\\nabla b_{e_i}(p) \\}$ is a global parallel ortho"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.00341","kind":"arxiv","version":3},"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":"2026-05-18T00:23:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HrQflHQiMsk1tBsvXGCwCvajTC9YnrHcLzTe4IL5KluqQOih8UmCuTgcKLmyq7QY586qxkuivpEUJwA+R8O4BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T14:26:52.687254Z"},"content_sha256":"c754b128ade3de613fdd6ed01eb1ba53a936557bebacdd2b6e12cb9a1358d16c","schema_version":"1.0","event_id":"sha256:c754b128ade3de613fdd6ed01eb1ba53a936557bebacdd2b6e12cb9a1358d16c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OE7KDZYJUCVG4HJQXNNCQ4KWQG/bundle.json","state_url":"https://pith.science/pith/OE7KDZYJUCVG4HJQXNNCQ4KWQG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OE7KDZYJUCVG4HJQXNNCQ4KWQG/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":"2026-06-27T14:26:52Z","links":{"resolver":"https://pith.science/pith/OE7KDZYJUCVG4HJQXNNCQ4KWQG","bundle":"https://pith.science/pith/OE7KDZYJUCVG4HJQXNNCQ4KWQG/bundle.json","state":"https://pith.science/pith/OE7KDZYJUCVG4HJQXNNCQ4KWQG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OE7KDZYJUCVG4HJQXNNCQ4KWQG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:OE7KDZYJUCVG4HJQXNNCQ4KWQG","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":"4a0c0778aede8ec7f6c92bd9eb526ee6ba9e5083beff88cfea94afc5ee0547da","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-03-01T15:20:49Z","title_canon_sha256":"331d3c65207821e89fce3ffd0fbb451421b3858a4c02866a0dbec7a820837009"},"schema_version":"1.0","source":{"id":"1703.00341","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1703.00341","created_at":"2026-05-18T00:23:09Z"},{"alias_kind":"arxiv_version","alias_value":"1703.00341v3","created_at":"2026-05-18T00:23:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.00341","created_at":"2026-05-18T00:23:09Z"},{"alias_kind":"pith_short_12","alias_value":"OE7KDZYJUCVG","created_at":"2026-05-18T12:31:34Z"},{"alias_kind":"pith_short_16","alias_value":"OE7KDZYJUCVG4HJQ","created_at":"2026-05-18T12:31:34Z"},{"alias_kind":"pith_short_8","alias_value":"OE7KDZYJ","created_at":"2026-05-18T12:31:34Z"}],"graph_snapshots":[{"event_id":"sha256:c754b128ade3de613fdd6ed01eb1ba53a936557bebacdd2b6e12cb9a1358d16c","target":"graph","created_at":"2026-05-18T00:23:09Z","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":"$(M^n,g)$ be a complete Riemannian manifold without conjugate points. In this paper, we show that if $M$ is also simply connected, then $M$ is flat, provided that $M$ is also asymptotically harmonic manifold with minimal horospheres (AHM). The (first order) flatness of $M$ is shown by using the strongest criterion: $\\{{e_i}\\}$ be an orthonormal basis of $T_{p}M$ and $\\{b_{e_{i}}\\}$ be the corresponding Busemann functions on $M$. Then, (1) The vector space $V = span\\{b_{v} | v \\in T_{p}M \\}$ is finite dimensional and dim $V = $ dim $M = n$.(2) $\\{\\nabla b_{e_i}(p) \\}$ is a global parallel ortho","authors_text":"Hemangi Shah","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-03-01T15:20:49Z","title":"Geometry of Asymptotically harmonic manifolds with minimal horospheres"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.00341","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:cce8b5b0e068e8945146a9c5d1bf289f61f62b1b66a00ae80b8ebaa5cf6947ef","target":"record","created_at":"2026-05-18T00:23:09Z","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":"4a0c0778aede8ec7f6c92bd9eb526ee6ba9e5083beff88cfea94afc5ee0547da","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-03-01T15:20:49Z","title_canon_sha256":"331d3c65207821e89fce3ffd0fbb451421b3858a4c02866a0dbec7a820837009"},"schema_version":"1.0","source":{"id":"1703.00341","kind":"arxiv","version":3}},"canonical_sha256":"713ea1e709a0aa6e1d30bb5a2871568191a9fa15d90ea87eee51501f68610fed","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"713ea1e709a0aa6e1d30bb5a2871568191a9fa15d90ea87eee51501f68610fed","first_computed_at":"2026-05-18T00:23:09.139851Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:23:09.139851Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LEBWhSxMBVDmiBOoxfqdNdRJwjWK+lxjXG5FamOuio9E7i58yAJ8hYuSY0/lLvvd+afFuhuLUXO/WCM9Zb9dDw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:23:09.140542Z","signed_message":"canonical_sha256_bytes"},"source_id":"1703.00341","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cce8b5b0e068e8945146a9c5d1bf289f61f62b1b66a00ae80b8ebaa5cf6947ef","sha256:c754b128ade3de613fdd6ed01eb1ba53a936557bebacdd2b6e12cb9a1358d16c"],"state_sha256":"9c50c56e48d415d9e7950b49ff7d4f18f629e684dc490d92bc2994526faccc0a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"O3pS2/3J+2lmmeRbRQ7efsUdIqQFRGjSvEX+R1WrxFREWmz/p4tF5IRp7HnhkdRd2NCrvVc6nlW6pFh80dwJDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T14:26:52.689110Z","bundle_sha256":"fd3d2b37ce405cf7b33e1e307dbbad85efdda088867b4fa6271edea666c97a86"}}