  We study the global structure of Lorentzian manifolds with partial sectional
curvature bounds. In particular, we prove completeness theorems for homogeneous
and isotropic cosmologies as well as static spherically symmetric spacetimes.
The latter result is used to rigorously prove the absence of static spherically
symmetric black holes in more than three dimensions. The proofs of these new
results are preceded by a detailed exposition of the local aspects of sectional
curvature bounds for Lorentzian manifolds, which extends and strengthens
previous constructions.
