{"paper":{"title":"Lagrangian immersions in the product of Lorentzian two manifold","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Nikos Georgiou","submitted_at":"2014-03-03T03:45:22Z","abstract_excerpt":"For Lorentzian 2-manifolds $(\\Sigma_1,g_1)$ and $(\\Sigma_2,g_2)$ we consider the two product para-K\\\"ahler structures $(G^{\\epsilon},J,\\Omega^{\\epsilon})$ defined on the product four manifold $\\Sigma_1\\times\\Sigma_2$, with $\\epsilon=\\pm 1$. We show that the metric $G^{\\epsilon}$ is locally conformally flat (resp. Einstein) if and only if the Gauss curvatures $\\kappa_1,\\kappa_2$ of $g_1,g_2$, respectively, are both constants satisfying $\\kappa_1=-\\epsilon\\kappa_2$ (resp. $\\kappa_1=\\epsilon\\kappa_2$). We give the conditions on the Gauss curvatures for which every Lagrangian surface with parallel"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.0305","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"}