  The role of curvature in relation with Lie algebra contractions of the
pseudo-ortogonal algebras so(p,q) is fully described by considering some
associated symmetrical homogeneous spaces of constant curvature within a
Cayley-Klein framework. We show that a given Lie algebra contraction can be
interpreted geometrically as the zero-curvature limit of some underlying
homogeneous space with constant curvature. In particular, we study in detail
the contraction process for the three classical Riemannian spaces (spherical,
Euclidean, hyperbolic), three non-relativistic (Newtonian) spacetimes and three
relativistic ((anti-)de Sitter and Minkowskian) spacetimes. Next, from a
different perspective, we make use of quantum deformations of Lie algebras in
order to construct a family of spaces of non-constant curvature that can be
interpreted as deformations of the above nine spaces. In this framework, the
quantum deformation parameter is identified as the parameter that controls the
curvature of such "quantum" spaces.
