  In [Heimann, Lehrenfeld, Preu{\ss}, SIAM J. Sci. Comp. 45(2), 2023, B139 -
B165] new geometrically unfitted space-time Finite Element methods for partial
differential equations posed on moving domains of higher-order accuracy in
space and time have been introduced. For geometrically higher-order accuracy a
parametric mapping on a background space-time tensor-product mesh has been
used. In this paper, we concentrate on the geometrical accuracy of the
approximation and derive rigorous bounds for the distance between the realized
and an ideal mapping in different norms and derive results for the space-time
regularity of the parametric mapping. These results are important and lay the
ground for the error analysis of corresponding unfitted space-time finite
element methods.
