Closed, negatively curved Riemannian manifolds with a flat horosphere are hyperbolic

2017/01/24 by Besson, Gérard, Courtois, Gilles, Hersonsky, Sa'ar
#Differential Geometry (math.DG) #FOS: Mathematics #Primary: 58B20 #Secondary 57N16

paper · doi:10.48550/arxiv.1701.06883

Abstract

We prove that the existence of one flat horosphere in the universal cover of a closed, strictly quarter pinched, negatively curved Riemannian manifold of dimension n with n greater than or equal to 3, implies that the manifold is homothetic to a real hyperbolic manifold.

Related

• Rigidity of flat holonomies
• On Horospherical Rigidity
• Horospheres in degenerate 3-manifolds
• Hyperbolic manifolds whose envelopes of holomorphy are not hyperbolic
• Hyperbolization and regular neighborhoods