{"paper":{"title":"Generalizing axioms of $r$-planes and $r$-spheres on Riemannian and K\\\"ahler manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Cristina Levina, S\\'ergio Mendon\\c{c}a","submitted_at":"2015-11-27T17:25:37Z","abstract_excerpt":"The famous theorems of Cartan, related to the axiom of $r$-planes, and Leung-Nomizu about the axiom of $r$-spheres were extended to K\\\"ahler geometry by several authors.\n  In this paper we replace the strong notions of totally geodesic submanifolds ($r$-planes) and extrinsic spheres ($r$-spheres) by a wider class of special isometric immersions such that theorems of type \"axioms of $r$-special submanifolds\" could hold. We verify also that there are plenty of special submanifolds in real and complex space forms and, in the codimension one case, in Einstein manifolds."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.08751","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"}