  Inflationary spacetimes have been argued to be past geodesically incomplete
in many situations. However, whether the geodesic incompleteness implies the
existence of an initial spacetime curvature singularity or whether the
spacetime may be extended (potentially into another phase of the universe) is
generally unknown. Both questions have important physical implications. In this
paper, we take a closer look at the geometrical structure of inflationary
spacetimes and investigate these very questions. We first classify which past
inflationary histories have a scalar curvature singularity and which might be
extendible and/or non-singular in homogeneous and isotropic cosmology with flat
spatial sections. Then, we derive rigorous extendibility criteria of various
regularity classes for quasi-de Sitter spacetimes that evolve from infinite
proper time in the past. Finally, we show that beyond homogeneity and isotropy,
special continuous extensions respecting the Einstein field equations with a
perfect fluid must have the equation of state of a de Sitter universe
asymptotically. An interpretation of our results is that past-eternal
inflationary scenarios are most likely physically singular, except in
situations with very special initial conditions.
