Ideal hypersurfaces of Euclidean four-space

TL;DR

This paper classifies and analyzes ideal hypersurfaces in Euclidean 4-space, focusing on their principal curvatures, rigidity properties, and providing explicit examples, advancing understanding of minimal and non-minimal cases.

Contribution

It classifies ideal hypersurfaces with two or three principal curvatures in Euclidean 4-space and investigates their rigidity, including explicit examples and non-rigidity cases.

Findings

Ideal hypersurfaces with two distinct principal curvatures are classified.

Such hypersurfaces are always rigid.

Non-minimal hypersurfaces with three principal curvatures are rigid.

Abstract

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.