  The notion of ideal immersions was introduced by the author in 1990s. Roughly
speaking, an ideal immersion of a Riemannian manifold into a real space form is
a nice isometric immersion which produces the least possible amount of tension
from the ambient space at each point.
  In this paper, we classify all ideal hypersurfaces with two distinct
principal curvatures in the Euclidean 4-space $\mathbb E^4$. Moreover, we prove
that such ideal hypersurfaces are always rigid. Furthermore, we show that
non-minimal ideal hypersurfaces with three distinct principal curvatures in
$\mathbb E^4$ are also rigid. On the other hand, we provide explicit examples
to illustrate that minimal ideal hypersurfaces with three principal curvatures
in $\mathbb E^4$ are not necessarily rigid.
