2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/222161We study geodesically complete and locally compact Hadamard spaces X whose Tits boundary is a connected irreducible spherical building. We show that X is symmetric iff complete geodesics in X do not branch and a Euclidean building otherwise. Furthermore, every boundary equivalence (cone topology homeomorphism preserving the Tits metric) between two such spaces is induced by a homothety. As an application, we can extend the Mostow and Prasad rigidity theorems to compact singular (orbi)spaces of nonpositive curvature which are homotopy equivalent to a quotient of a symmetric space or Euclidean building by a cocompact group of isometries.My 1997 habilitation thesis as published in Bonner Mathematische Schriften vol 326 (2000)Metric GeometryDifferential Geometry51E24; 53C35; 53C24; 51K10text