Effects of Spatial Curvature on the $f(R)$ Gravity Phase Space: no
  Inflationary Attractor?
  In this paper we study the effects of spatial curvature of the metric on the
phase space of vacuum $f(R)$ gravity. Particularly, we appropriately choose the
variables of the dynamical system, in order for this to be autonomous, and we
study the phase space of the resulting theory, focusing on de Sitter, matter
and radiation domination fixed points. Our analysis indicates that the effect
of spatial curvature on the phase space is radical, since it destabilizes all
the stable de Sitter vacua of the flat spacetime vacuum $f(R)$ gravity phase
space, making the phase space having non-trivial unstable submanifolds. This
instability occurs regardless if the spacetime has elliptic or hyperbolic
spatial sections, and it is also robust towards the choice of initial
conditions. We investigate the source of the instability in the system, and
also we discuss the stability of the matter and radiation domination vacua,
which, as we demonstrate, are also highly unstable. Our results for de Sitter
attractors indicate that the stable de Sitter attractors of the vacuum $f(R)$
gravity theory for a flat Universe, are destabilized by the presence of
curvature, and this shows that inflation for vacuum $f(R)$ gravity in non-flat
spacetime is problematic, at least at the phase space level. This result holds
true for both elliptic and hyperbolic spacetimes.
