{"paper":{"title":"Hypersurfaces of Product Spaces with a Canonical Direction","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Pedro Roitman, Ronaldo F. de Lima","submitted_at":"2019-05-28T18:45:48Z","abstract_excerpt":"Consider a complete Riemannian manifold $M^n$ and let $\\Sigma^n$ be an orientable hypersurface of the product manifold $M\\times\\mathbb{R}$ endowed with its standard product metric $\\langle \\,,\\, \\rangle.$ Let $\\nabla\\xi$ denote the gradient of the height function $\\xi$ of $\\Sigma.$ In this note, we characterize the hypersurfaces $\\Sigma$ which have $\\nabla\\xi$ as a principal direction. Our approach is based on the work of R. Tojeiro, who considered the case where $M$ is a constant sectional curvature space form."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.12026","kind":"arxiv","version":2},"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"}