Local and infinitesimal rigidity of simply connected negatively curved manifols

2015/07/17 by Biswas, Kingshook
#53C24 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1507.04974

Abstract

Let (X, g₀) be a simply connected, complete, negatively curved Riemannian manifold. We prove local and infinitesimal rigidity results for compactly supported deformations of the metric g₀. For any negatively curved metric g equal to g₀ outside a compact, the identity map of X induces a natural boundary map between the boundaries at infinity of X with respect to g₀ and g. We show that if (g_t) is a smooth 1-parameter family of negatively curved metrics all equal to g₀ outside a fixed compact then if all the boundary maps (between the boundaries of X with respect to g₀ and g_t) are Moebius then the metrics g_t are all isometric to g₀. We also show that given a compact K in X, there is a neighbourhood of g₀ in the C^2,α topology such that for any negatively curved metric g in this neighbourhood which is equal to g₀ outside K, if the boundary map is Moebius and the g₀ and g volumes of K agree then g is isometric to g₀.

Related

• Moebius rigidity for compact deformations of negatively curved manifolds
• Moebius rigidity for simply connected, negatively curved surfaces
• Local lens rigidity for manifolds of Anosov type
• Rigidity of Totally Geodesic Hypersurfaces in Negative Curvature
• Smooth Volume Rigidity for Manifolds with Negatively Curved Targets