  We examine the effects of dark energy dynamics and spatial curvature on
cosmic reionization by studying reionization in tilted spatially-flat and
untilted non-flat XCDM and $\phi$CDM dynamical dark energy inflation models
that best fit the Planck 2015 cosmic microwave background (CMB) anisotropy and
a large compilation of non-CMB data. We carry out a detailed statistical study,
based on a principal component analysis and a Markov chain Monte Carlo analysis
of a compilation of lower-redshift reionization data, to estimate the
uncertainties in the cosmological model reionization histories. We find that,
irrespective of the nature of dark energy, there are significant differences
between the reionization histories of the spatially-flat and non-flat models.
Although both the flat and non-flat models can accurately match the
low-redshift ($z\lesssim6$) reionization observations, there is a clear
discrepancy between high-redshift ($z>7$) Lyman-$\alpha$ emitter data and the
predictions from non-flat models. This is solely due to the fact that the
non-flat models have a significantly larger electron scattering optical depth,
$\tau_{\rm el}$, compared to the flat models, which requires an extended and
much earlier reionization scenario supported by more high-redshift ionizing
sources in the non-flat models. Non-flat models also require strong redshift
evolution in the photon escape fraction, that can become unrealistically high
($\gtrsim1$) at some redshifts. However, $\tau_{\rm el}$ is about 0.9-$\sigma$
lower in the tilted flat $\Lambda$CDM model when the new Planck 2018 data are
used and this reduction will partially alleviate the tension between the
non-flat model predictions and the data.
