  We define \emph{piecewise rank 1} manifolds, which are aspherical manifolds
that generally do not admit a nonpositively curved metric but can be decomposed
into pieces that are diffeomorphic to finite volume, irreducible, locally
symmetric, nonpositively curved manifolds with $\pi_1$-injective cusps. We
prove smooth (self) rigidity for this class of manifolds in the case where the
gluing preserves the cusps' homogeneous structure. We compute the group of self
homotopy equivalences of such a manifold and show that it can contain a normal
free abelian subgroup and thus, can be infinite. Elements of this abelian
subgroup are twists along elements in the center of the fundamental group of a
cusp.
