We will report the study of certain rigidity phenomena onthe moduli space of metrics with non-positive sectional curvature ona compact manifold: if there is one metric with a compatible localsplitting structure, then every metric has a compatible local splittingstructure. Compatible local splitting structures has been found oncollapsed manifolds with bounded non-positive sectional curvature.This is a joint program with Jianguo Cao and Jeff Cheeger.