On ideal hypersurfaces of Euclidean 4-space

The notion of ideal immersions was introduced by the author in the 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 E4E4. Moreover, we prove that such ideal hypersurfaces are always rigid. Furthermore, we show that non-minimal ideal hypersurfaces with three distinct principal curvatures in E4E4 are also rigid. On the other hand, we provide explicit examples to illustrate that minimal ideal hypersurfaces with three principal curvatures in E4E4 are not necessarily rigid.