In this paper we give a new definition of harmonic curvature functions in terms of B2 and we define a new kind of slant helix which we call quaternionic B2−slant helix in 4−dimensional Euclidean space E 4 by using the new harmonic curvature functions. Also we define a vector field D which we call Darboux quaternion of the real quaternionic B2−slant helix in 4−dimensional Euclidean space E 4 and we give a new characterization such as:where H2, H1 are harmonic curvature functions and K is the principal curvature function of the curve α.

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We study and examine the rotational hypersurface and its Gauss map in Euclidean four-space 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 E 4 . We also draw some figures of the rotational hypersurface.

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