The semi-rigidity of nonpositively curved metrics
Abstract. We call the moduli space of non-positively curved metrics on a closed manifold rigid if all the metrics are isometric up to a rescaling. The moduli space is rigid if it contains an irreducible metric of rank at least two (the higher rank rigidity).
We call a moduli space semi-rigid, if all metrics are `alike'
in the following sense: each metric poses a certain compatible
local splitting structure whose underlying topological structure
is independent of the metric. A typical example is the moduli
space of nonpositively curved metrics on a graph 
-manifold.
The semi-rigidity is closely related to the collapsing theory
of Cheeger-Gromov. In this talk, we will discuss the semi-rigidity
and its applications.