We study Einstein warped product spaces. As a result, we prove the following: if $M$ is an Einstein warped product space with nonpositive scalar curvature and compact base, then $M$ is simply a Riemannian product space.

We study and examine the rotational hypersurface and its Gauss map in Euclidean four-space ${\mathbb{E}}^4$ . We calculate the Gauss map, the mean curvature and the Gaussian curvature of the rotational hypersurface and obtain some results. Then, we introduce the third Laplace–Beltrami operator. Moreover, we calculate the third Laplace–Beltrami operator of the rotational hypersurface in ${\mathbb{E}}^4.$ We also draw some figures of the rotational hypersurface.

The family of cones is one of typical models of non-cylindrical ruled surfaces. Among them, the circular cones are unique in the sense that their Gauss map satisfies a partial differential equation similar, though not identical, to one characterizing the

In the present article we study the rotational embedded surfaces in E^4. The rotational embedded surface was first studied by G. Ganchev and V. Milousheva as a surface in E^4. The Otsuki (non-round) sphere in E^4 is one of the special examples of this surface. Finally, we give necessary and sufficient conditions for the flat Ganchev-Milousheva rotational surface to have pointwise 1-type Gauss map.