H-hypersurfaces with three distinct principal curvatures in the Euclidean spaces

Abstract

The aim of the present paper is to characterize H-hypersurfaces with three distinct principal curvatures in the Euclidean space. A hypersurface with non-constant first mean curvature is said to be a H-hypersurface if it satisfies \(S(\nabla{s}_{1})=-\frac{s_{1}}{2}\nabla s_{1},\) where \(S\) is the shape operator and \(s_{1}\) denotes the first mean curvature of the hypersurface \(M\). At first the author deduces the connection forms of H-hypersurfaces and proves some lemmas. With the help of two lemmas he proves an interesting result. Then he obtains necessary and sufficient conditions for a hypersurface \(M\) in \(E^{n+1}\) to be a H-hypersurface. Finally, the geometric interpretation of the obtained results is explained. The results of the paper are very interesting and the quality of the paper will enrich the field of hypersurfaces in Euclidean space.