{"paper":{"title":"Geometry of Asymptotically harmonic manifolds with minimal horospheres","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Hemangi Shah","submitted_at":"2017-03-01T15:20:49Z","abstract_excerpt":"$(M^n,g)$ be a complete Riemannian manifold without conjugate points. In this paper, we show that if $M$ is also simply connected, then $M$ is flat, provided that $M$ is also asymptotically harmonic manifold with minimal horospheres (AHM). The (first order) flatness of $M$ is shown by using the strongest criterion: $\\{{e_i}\\}$ be an orthonormal basis of $T_{p}M$ and $\\{b_{e_{i}}\\}$ be the corresponding Busemann functions on $M$. Then, (1) The vector space $V = span\\{b_{v} | v \\in T_{p}M \\}$ is finite dimensional and dim $V = $ dim $M = n$.(2) $\\{\\nabla b_{e_i}(p) \\}$ is a global parallel ortho"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.00341","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"}