2008/12/17 by Jianguo Cao, Jianguo Caoand, Xiaoyang Chen +2
Mathematics · #53C20 #53C42 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities #math.DG #msc:53C20 #msc:53C42
paper · pdf · doi:10.48550/arxiv.0812.3353
In this updated version, we were able to extend our results on smooth visibility manifolds to possibly singular visibility spaces by using a result of Martin Bridson
openalex publication_date 2008/12/17 · arxiv created 2009/02/16 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
Theorem A. Let Mn denote a closed Riemannian manifold with nonpositive sectional curvature and let Mn be the universal cover of Mn with the lifted metric. Suppose that the universal cover Mn contains no totally geodesic embedded Euclidean plane ℝ2 (i.e., Mn is a visibility manifold). Then Gromov's simplicial volume ‖ Mn ‖ is non-zero. Consequently, Mn is non-collapsible while keeping Ricci curvature bounded from below. More precisely, if Ricg ≥ -(n-1), then vol(Mn, g) ≥ (1)/((n-1)n n!) ‖ Mn ‖ > 0. Theorem B. (Perelman) Let M3 be a closed a-spherical 3-manifold (K(π, 1)-space) with the fundamental group Γ. Suppose that Γ contains no subgroups isomorphic to ℤ⊕ ℤ. Then M3 is diffeomorphic to a compact quotient of real hyperbolic space ℍ3, i.e., M3 ≡ ℍ3/Γ. Consequently, MinVol(M3) ≥ 1/24‖ M3 ‖ > 0. Minimal volume and simplicial norm of all other compact 3-manifolds without boundary and \it singular spaces will also be discussed.