{"paper":{"title":"On Some Basic Results Related to Affine Functions on Riemmanian Manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Chong Li, Jen-Chih Yao, Xiangmei Wang","submitted_at":"2015-09-24T07:42:22Z","abstract_excerpt":"We study some basic properties of the function $f_0:M\\rightarrow\\IR$ on Hadamard manifolds defined by $$ f_0(x):=\\langle u_0,\\exp_{x_0}^{-1}x\\rangle\\quad\\mbox{for any $x\\in M$}. $$ A characterization for the function to be linear affine is given and a counterexample on Poincar\\'e plane is provided, which in particular, shows that assertions (i) and (ii) claimed in \\cite[Proposition 3.4]{Papa2009} are not true, and that the function $f_0$ is indeed not quasi-convex. Furthermore, we discuss the convexity properties of the sub-level sets of the function on Riemannian manifolds with constant secti"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.07264","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"}