{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:W7CPAIFBYFGMO3CSAOFQND44TL","short_pith_number":"pith:W7CPAIFB","schema_version":"1.0","canonical_sha256":"b7c4f020a1c14cc76c52038b068f9c9afd7189ca78c8f58fc32657f0b7a36151","source":{"kind":"arxiv","id":"1207.1944","version":1},"attestation_state":"computed","paper":{"title":"On Einstein Kropina metrics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Xiaoling Zhang, Yi-Bing Shen","submitted_at":"2012-07-09T04:35:23Z","abstract_excerpt":"In this paper, a characteristic condition of Einstein Kropina metrics is given. By the characteristic condition, we prove that a non-Riemannian Kropina metric $F=\\frac{\\alpha^2}{\\beta}$ with constant Killing form $\\beta$ on an n-dimensional manifold $M$, $n\\geq 2$, is an Einstein metric if and only if $\\alpha$ is also an Einstein metric. By using the navigation data $(h,W)$, it is proved that an n-dimensional ($n\\geq2$) Kropina metric $F=\\frac{\\alpha^2}{\\beta}$ is Einstein if and only if the Riemannian metric $h$ is Einstein and $W$ is a unit Killing vector field with respect to $h$. Moreover,"},"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":"1207.1944","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-07-09T04:35:23Z","cross_cats_sorted":[],"title_canon_sha256":"d99871bf44efa1da738489e35ab9eaa3b134e9d86ef01b6165a6747d9878e3b9","abstract_canon_sha256":"c0d81aa24ba3439dbf5420a58e954478685f25cd7137f8b206a58945d08e53e6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:51:26.542884Z","signature_b64":"2VN1Z40/SkbrMSJE6z0UWyWgZtWjbUFhOaR8KDaHPdYAx9VhDrKrmh9Hu/dD1Yf8esbEvd3vYgqt87/XK0MrDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b7c4f020a1c14cc76c52038b068f9c9afd7189ca78c8f58fc32657f0b7a36151","last_reissued_at":"2026-05-18T03:51:26.542358Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:51:26.542358Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Einstein Kropina metrics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Xiaoling Zhang, Yi-Bing Shen","submitted_at":"2012-07-09T04:35:23Z","abstract_excerpt":"In this paper, a characteristic condition of Einstein Kropina metrics is given. By the characteristic condition, we prove that a non-Riemannian Kropina metric $F=\\frac{\\alpha^2}{\\beta}$ with constant Killing form $\\beta$ on an n-dimensional manifold $M$, $n\\geq 2$, is an Einstein metric if and only if $\\alpha$ is also an Einstein metric. By using the navigation data $(h,W)$, it is proved that an n-dimensional ($n\\geq2$) Kropina metric $F=\\frac{\\alpha^2}{\\beta}$ is Einstein if and only if the Riemannian metric $h$ is Einstein and $W$ is a unit Killing vector field with respect to $h$. Moreover,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.1944","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":"1207.1944","created_at":"2026-05-18T03:51:26.542433+00:00"},{"alias_kind":"arxiv_version","alias_value":"1207.1944v1","created_at":"2026-05-18T03:51:26.542433+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.1944","created_at":"2026-05-18T03:51:26.542433+00:00"},{"alias_kind":"pith_short_12","alias_value":"W7CPAIFBYFGM","created_at":"2026-05-18T12:27:25.539911+00:00"},{"alias_kind":"pith_short_16","alias_value":"W7CPAIFBYFGMO3CS","created_at":"2026-05-18T12:27:25.539911+00:00"},{"alias_kind":"pith_short_8","alias_value":"W7CPAIFB","created_at":"2026-05-18T12:27:25.539911+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/W7CPAIFBYFGMO3CSAOFQND44TL","json":"https://pith.science/pith/W7CPAIFBYFGMO3CSAOFQND44TL.json","graph_json":"https://pith.science/api/pith-number/W7CPAIFBYFGMO3CSAOFQND44TL/graph.json","events_json":"https://pith.science/api/pith-number/W7CPAIFBYFGMO3CSAOFQND44TL/events.json","paper":"https://pith.science/paper/W7CPAIFB"},"agent_actions":{"view_html":"https://pith.science/pith/W7CPAIFBYFGMO3CSAOFQND44TL","download_json":"https://pith.science/pith/W7CPAIFBYFGMO3CSAOFQND44TL.json","view_paper":"https://pith.science/paper/W7CPAIFB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1207.1944&json=true","fetch_graph":"https://pith.science/api/pith-number/W7CPAIFBYFGMO3CSAOFQND44TL/graph.json","fetch_events":"https://pith.science/api/pith-number/W7CPAIFBYFGMO3CSAOFQND44TL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/W7CPAIFBYFGMO3CSAOFQND44TL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/W7CPAIFBYFGMO3CSAOFQND44TL/action/storage_attestation","attest_author":"https://pith.science/pith/W7CPAIFBYFGMO3CSAOFQND44TL/action/author_attestation","sign_citation":"https://pith.science/pith/W7CPAIFBYFGMO3CSAOFQND44TL/action/citation_signature","submit_replication":"https://pith.science/pith/W7CPAIFBYFGMO3CSAOFQND44TL/action/replication_record"}},"created_at":"2026-05-18T03:51:26.542433+00:00","updated_at":"2026-05-18T03:51:26.542433+00:00"}