  We establish Korn's interpolation inequalities and the rigidity results of
the strain tensor of the middle surface for the parabolic and elliptic shells
and show that the best constant in Korn's inequalities scales like $h^{3/2}$
for the parabolic shell and $h$ for the elliptic shell, removing the main
assumption that the middle surface of the shell is given by one single
principal coordinate in the literature and, in particular, including the closed
elliptic shell.
