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Some relations involving ${\\cal T}_{a}$-curvature tensor in $(N(k),\\xi)$-semi-Riemannian manifolds are proved. $\\xi $-${\\cal T}_{a}$-flat $(N(k),\\xi)$-semi-Riemannian manifolds are defined. It is proved that if $M$ is an $n$-dimensional $\\xi $-${\\cal T}_{a}$-flat $(N(k),\\xi)$-semi-Riemannian manifold, then it is $\\eta $-Einstein under an algebraic condition. We prove that a semi-Riemannian manifold, which is $T$-recurrent or $T$-symmetric, is always $T$-semisymmetric"},"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":"1202.6138","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-02-28T07:26:16Z","cross_cats_sorted":[],"title_canon_sha256":"23ba2fc3e411957c72447dca35b867bab362b44c1f1d5fd3c35160f8491ab3cc","abstract_canon_sha256":"63ed42289380af65613752856d2c4b3f35426e8b2d06b61e72d70746134206ef"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:01:14.122417Z","signature_b64":"Hrk4EqUAXAOhc6RShtGzek8xY/GrnwFo1aXQaPPChe4NFFlthyUtOOdapFJ1caYI2tmjlPC32WdLYaNII7oSCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a5ff10371f54a68bbaf66cfa0a565325d0ecaec38eaa7ea8500959b0d4a3c21d","last_reissued_at":"2026-05-18T04:01:14.121988Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:01:14.121988Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On $(N(k),\\xi)$-semi-Riemannian manifolds: Semisymmetries","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Mukut Mani Tripathi, Punam Gupta","submitted_at":"2012-02-28T07:26:16Z","abstract_excerpt":"$(N(k),\\xi)$-semi-Riemannian manifolds are defined. Examples and properties of $(N(k),\\xi)$-semi-Riemannian manifolds are given. Some relations involving ${\\cal T}_{a}$-curvature tensor in $(N(k),\\xi)$-semi-Riemannian manifolds are proved. $\\xi $-${\\cal T}_{a}$-flat $(N(k),\\xi)$-semi-Riemannian manifolds are defined. It is proved that if $M$ is an $n$-dimensional $\\xi $-${\\cal T}_{a}$-flat $(N(k),\\xi)$-semi-Riemannian manifold, then it is $\\eta $-Einstein under an algebraic condition. 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