Cosmologies with turning points

We explore singularity-free and geodesically-complete cosmologies based on manifolds that are not quite Lorentzian. The metric can be either smooth everywhere or non-degenerate everywhere, but both, depending the coordinate system. gives an Einstein tensor is first order in derivatives while has a piecewise FLRW form. On such manifold universe transition from expanding to contracting, vice versa, with equations satisfied without violation of standard energy conditions. also obtain corresponding extension Kasner vacuum solutions manifolds.