{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:WHJ5X4PXCTBKP46TCBUS35LXBE","short_pith_number":"pith:WHJ5X4PX","schema_version":"1.0","canonical_sha256":"b1d3dbf1f714c2a7f3d310692df577093f75fe60ca65273ef0c451c6c7b7a80f","source":{"kind":"arxiv","id":"1702.03646","version":1},"attestation_state":"computed","paper":{"title":"Hessian of Busemann functions and rank of Hadamard manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Hiroyasu Satoh, JeongHyeong Park, Mitsuhiro Itoh, Sinwhi Kim","submitted_at":"2017-02-13T06:20:11Z","abstract_excerpt":"In this article we show that every geodesic is rank one and the Hessian of Busemann functions is positive definite for a harmonic Damek-Ricci space, a two step solvable Lie group with a left invariant metric. Moreover, the eigenspace of the Hessian of Busemann functions on a Hadamard manifold $(M,g)$ corresponding to eigenvalue zero is investigated with respect to rank of geodesics. On a harmonic Hadamard manifold which is of purely exponential volume growth, or of hypergeometric type it is shown that every Busemann function admits positive definite Hessian. A criterion for $(M,g)$ fulfilling "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1702.03646","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-02-13T06:20:11Z","cross_cats_sorted":[],"title_canon_sha256":"b4bc08b82a62d845c5608e426699376aa7a5d8596054d888e51881f6a5c0d173","abstract_canon_sha256":"3523e82c7a55c2e6c18f0e4070d4ff0feaab8afc9b6f61556906e32dbd6bf04a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:50:53.299136Z","signature_b64":"RcMB6WnRIv/nxnoII8IatcbtsAo7XUIlBjvlE7TKduwh5WbAS+pYz2rhx7GtTBA2v7nSoz8qZBKbFmZCmm5oCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b1d3dbf1f714c2a7f3d310692df577093f75fe60ca65273ef0c451c6c7b7a80f","last_reissued_at":"2026-05-18T00:50:53.298576Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:50:53.298576Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hessian of Busemann functions and rank of Hadamard manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Hiroyasu Satoh, JeongHyeong Park, Mitsuhiro Itoh, Sinwhi Kim","submitted_at":"2017-02-13T06:20:11Z","abstract_excerpt":"In this article we show that every geodesic is rank one and the Hessian of Busemann functions is positive definite for a harmonic Damek-Ricci space, a two step solvable Lie group with a left invariant metric. Moreover, the eigenspace of the Hessian of Busemann functions on a Hadamard manifold $(M,g)$ corresponding to eigenvalue zero is investigated with respect to rank of geodesics. On a harmonic Hadamard manifold which is of purely exponential volume growth, or of hypergeometric type it is shown that every Busemann function admits positive definite Hessian. A criterion for $(M,g)$ fulfilling "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.03646","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1702.03646","created_at":"2026-05-18T00:50:53.298639+00:00"},{"alias_kind":"arxiv_version","alias_value":"1702.03646v1","created_at":"2026-05-18T00:50:53.298639+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1702.03646","created_at":"2026-05-18T00:50:53.298639+00:00"},{"alias_kind":"pith_short_12","alias_value":"WHJ5X4PXCTBK","created_at":"2026-05-18T12:31:53.515858+00:00"},{"alias_kind":"pith_short_16","alias_value":"WHJ5X4PXCTBKP46T","created_at":"2026-05-18T12:31:53.515858+00:00"},{"alias_kind":"pith_short_8","alias_value":"WHJ5X4PX","created_at":"2026-05-18T12:31:53.515858+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/WHJ5X4PXCTBKP46TCBUS35LXBE","json":"https://pith.science/pith/WHJ5X4PXCTBKP46TCBUS35LXBE.json","graph_json":"https://pith.science/api/pith-number/WHJ5X4PXCTBKP46TCBUS35LXBE/graph.json","events_json":"https://pith.science/api/pith-number/WHJ5X4PXCTBKP46TCBUS35LXBE/events.json","paper":"https://pith.science/paper/WHJ5X4PX"},"agent_actions":{"view_html":"https://pith.science/pith/WHJ5X4PXCTBKP46TCBUS35LXBE","download_json":"https://pith.science/pith/WHJ5X4PXCTBKP46TCBUS35LXBE.json","view_paper":"https://pith.science/paper/WHJ5X4PX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1702.03646&json=true","fetch_graph":"https://pith.science/api/pith-number/WHJ5X4PXCTBKP46TCBUS35LXBE/graph.json","fetch_events":"https://pith.science/api/pith-number/WHJ5X4PXCTBKP46TCBUS35LXBE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WHJ5X4PXCTBKP46TCBUS35LXBE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WHJ5X4PXCTBKP46TCBUS35LXBE/action/storage_attestation","attest_author":"https://pith.science/pith/WHJ5X4PXCTBKP46TCBUS35LXBE/action/author_attestation","sign_citation":"https://pith.science/pith/WHJ5X4PXCTBKP46TCBUS35LXBE/action/citation_signature","submit_replication":"https://pith.science/pith/WHJ5X4PXCTBKP46TCBUS35LXBE/action/replication_record"}},"created_at":"2026-05-18T00:50:53.298639+00:00","updated_at":"2026-05-18T00:50:53.298639+00:00"}