2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/77348Under mild assumptions on a group G, we prove that the class of complete Riemannian n-manifolds of uniformly bounded negative sectional curvatures and with the fundamental groups isomorphic to G breaks into finitely many tangential homotopy types. It follows that many aspherical manifolds do not admit complete negatively curved metrics with prescribed curvature bounds.22 pages, no figures; to appear in Journal of Differential GeometryDifferential GeometryGroup TheoryGeometric TopologyMetric Geometry53C23; 20F32; 55R50; 57S30text