Local properties and global structure of McVittie spacetimes with
  non-flat FLRW backgrounds
  McVittie spacetimes embed the vacuum Schwarzschild(-(anti) de Sitter)
spacetime in an isotropic FLRW background universe. We study the global
structure of McVittie spacetimes with spatially non-flat FLRW backgrounds. This
requires the extension of the definition of such spacetimes, previously given
only for the flat and open cases, to the closed case. We revisit this
definition and show how it gives rise to a unique spacetime (given the FLRW
background, the mass parameter $M$ and the cosmological constant $\Lambda$) in
the open and flat cases. In the closed case, an additional free function of the
cosmic time arises. We derive some basic results on the metric, curvature and
matter content of McVittie spacetimes and derive a representation of the line
element that makes the study of their global properties possible. In the closed
case (independently of the free function mentioned above), the spacetime is
confined (at each instant of time) to a region bounded by a minimum and a
maximum area radius, and is bounded either to the future or to the past by a
scalar curvature singularity. This allowed region only exists when the
background scale factor is above a certain minimum. In the open case, radial
null geodesics originate in finite affine time in the past at a boundary formed
by the union of the Big Bang singularity of the FLRW background and a
non-singular hypersurface of varying causal character. Furthermore, in the case
of eternally expanding open universes, we show that black holes are ubiquitous:
ingoing radial null geodesics extend in finite affine time to a hypersurface
that forms the boundary of the region from which photons can escape to future
null infinity. We revisit the black hole interpretation of McVittie spacetimes
in the spatially flat case, and show that this interpretation holds also in the
case of a vanishing cosmological constant, contrary to a previous claim of
ours.
