{"paper":{"title":"An Isometrical ${\\Bbb C\\Bbb P}^{n}$-Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Hongwei Sun, Xiaole Su, Yusheng Wang","submitted_at":"2015-06-11T03:11:53Z","abstract_excerpt":"Let $M^n\\ (n\\geq3)$ be a complete Riemannian manifold with $\\sec_M\\geq 1$, and let $M_i^{n_i}$ ($i=1,2$) be two comlplete totally geodesic submanifolds in $M$. We prove that if $n_1+n_2=n-2$ and if the distance $|M_1M_2|\\geq\\frac{\\pi}{2}$, then $M_i$ is isometric to $\\Bbb S^{n_i}/\\Bbb Z_h$, ${\\Bbb C\\Bbb P}^{\\frac {n_i}2}$ or ${\\Bbb C\\Bbb P}^{\\frac {n_i}2}/\\Bbb Z_2$ with the canonical metric when $n_i>0$, and thus $M$ is isometric to $\\Bbb S^n/\\Bbb Z_h$, ${\\Bbb C\\Bbb P}^{\\frac n2}$ or ${\\Bbb C\\Bbb P}^{\\frac n2}/\\Bbb Z_2$ except possibly when $n=3$ and $M_1$ (or $M_2$) $\\stackrel{\\rm iso}{\\cong}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.03535","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"}