  We study in this paper three natural notions of convergence of homogeneous
manifolds, namely infinitesimal, local and pointed, and their relationship with
a fourth one, which only takes into account the underlying algebraic structure
of the homogeneous manifold and is indeed much more tractable. Along the way,
we introduce a subset of the variety of Lie algebras which parameterizes the
space of all n-dimensional simply connected homogeneous spaces with
q-dimensional isotropy, providing a framework which is very advantageous to
approach variational problems for curvature functionals as well as geometric
evolution equations on homogeneous manifolds.
