2008/01/01 by Harish Seshadri, Seshadri, Harish
Engineering · Mathematics · #Differential Geometry (math.DG) #Elasticity and Material Modeling #FOS: Mathematics #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.0801.0285
openalex publication_date 2008/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Using elementary comparison geometry, we prove: Let (M,g) be a simply-connected complete Riemannian manifold of dimension ≥ 3. Suppose that the sectional curvature K satisfies -1-s(r) ≤ K ≤ -1, where r denotes distance to a fixed point in M. If limr \rt ∞ e2rs(r) =0, then (M,g) has to be isometric to \mathbb Hn. The same proof also yields that if K satisfies -s(r) ≤ K ≤ 0 where limr \rt ∞ r2s(r)=0, then (M,g) is isometric to \Rn, a result due to Greene and Wu. Our second result is a local one: Let (M,g) be any Riemannian manifold. For a ∈ \R, if K ≤ a on a geodesic ball Bp(R) in M and K = a on ∂ Bp(R), then K= a on Bp(R).