  In this paper, as a continuation of \cite{YZ:inrdius},we develop the geometry
of the limit spaces of compact Riemannian manifolds with boundary, where we
assume a lower sectional curvature bound,two sides bounds on the second
fundamental forms of boundaries and an upper diameter bound. We mainly focus on
the general case of non inradius collapse/convergence, where inradii of
manifolds are uniformly bounded away from zero. In this case, many limit spaces
have wild geometry, which arise from the boundary behavior of
manifolds.Therefore the study of boundary singular points is a key to
understand such limit spaces. We also present some global
convergence/collapsing results.
