  Methods and properties regarding the linear perturbations are discussed for
some spatially closed (vacuum) solutions of Einstein's equation. The main focus
is on two kinds of spatially locally homogeneous solution; one is the Bianchi
III (Thurston's H^2 x R) type, while the other is the Bianchi II (Thurston's
Nil) type. With a brief summary of previous results on the Bianchi III
perturbations, asymptotic solutions for the gauge-invariant variables for the
Bianchi III are shown, with which (in)stability of the background solution is
also examined. The issue of linear stability for a Bianchi II solution is still
an open problem. To approach it, appropriate eigenfunctions are presented for
an explicitly compactified Bianchi II manifold and based on that, some field
equations on the Bianchi II background spacetime are studied. Differences
between perturbation analyses for Bianchi class B (to which Bianchi III
belongs) and class A (to which Bianchi II belongs) are stressed for an
intention to be helpful for applications to other models.
