{"paper":{"title":"Submanifolds of products of space forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Bruno Mendon\\c{c}a, Ruy Tojeiro","submitted_at":"2012-03-16T19:33:07Z","abstract_excerpt":"We give a complete classification of submanifolds with parallel second fundamental form of a product of two space forms. We also reduce the classification of umbilical submanifolds with dimension $m\\geq 3$ of a product $\\Q_{k_1}^{n_1}\\times \\Q_{k_2}^{n_2}$ of two space forms whose curvatures satisfy $k_1+k_2\\neq 0$ to the classification of $m$-dimensional umbilical submanifolds of codimension two of $\\Sf^n\\times \\R$ and $\\Hy^n\\times \\R$. The case of $\\Sf^n\\times \\R$ was carried out in \\cite{mt}. As a main tool we derive reduction of codimension theorems of independent interest for submanifolds"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.3790","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"}