Let M and M′ be simply
connected, complete Riemannian manifolds of nonpositive sectional curvature and let
Γ and Γ′ be properly discontinuous groups of isometries acting freely on M and
M′ respectively such that M∕Γ and M′∕Γ′ are compact. Let 𝜃 : Γ → Γ′ be
an isomorphism. There exists a pseudo-isometry ϕ : M → M′ such that
ϕ(γx) = 𝜃(γ)ϕ(x) for all γ in Γ and x in M. The question is whether this
pseudo-isometry ϕ can be extended to a homeomorphism ϕ between the boundaries
M(∞) and M′(∞) of M and M′ respective]y. This homeomorphism is further
required to be equivariant with respect to the isomorphism 𝜃. This extendability is
called the weak rigidity of compact nonpositively curved manifolds. In this
paper, this weak rigidity question is answered affirmatively if M is a simply
connected, complete Riemannian manifold of negative sectional curvature
and M′ is a noncompact symmetric space of rank one. If M and M′ are
noncompact symmetric spaces without direct factors of closed one or two
dimensional geodesic subspaces, then this weak rigidity is proved by G. D. Mostow
and is a part of his important strong rigidity theory of compact, locally
symmetric Riemannian manifolds. This paper is motivated by this theory of
Mostow.
