Vol. 78, No. 2, 1978

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ISSN: 0030-8730
Weak rigidity of compact negatively curved manifolds

Su-Shing Chen

Vol. 78 (1978), No. 2, 273–278
DOI: 10.2140/pjm.1978.78.273
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Let M and Mbe simply connected, complete Riemannian manifolds of nonpositive sectional curvature and let Γ and Γbe properly discontinuous groups of isometries acting freely on M and Mrespectively such that M∕Γ and MΓare compact. Let θ : Γ Γbe an isomorphism. There exists a pseudo-isometry ϕ : M Msuch 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 Mrespective]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 Mis a noncompact symmetric space of rank one. If M and Mare 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.

Mathematical Subject Classification 2000
Primary: 53C30
Secondary: 22E40
Received: 15 August 1977
Published: 1 October 1978
Su-Shing Chen