Two Hypersurfaces With Mutual Gauss Map And Translation Hypersurfaces Of Hypersurfaces In Sⁿ⁺¹

In this paper, we first give the notation of higher-order mean curvature， then studythe relations between the higher-order mean curvatures, Riemann curvatures, principalcurvatures of two Hypersurfaces which have property: one is the Gauss map of the otherin Sn+1. In particular, we research generalized rotation Hypersurfaces in Sn+1, andwe show that for a (k+2) stype rotation, it’s Gauss mapping is still a (k+2) styperotation. As an example, we show that the W illmore torus and Clifford torus is themutual of Guass mapping, and because Clifford torus is minimal surface, W illmoretorus is n minimal. Finally we study The general situation, that is the ralations betweenthe principal curvature of the Hypersurfaces in Sn+1and its translation Hypersurfaces,and we get the conditions which are satisfied by the principle curvature of Hypersurfacesin Sn+1, if we translate Hypersurfaces in Sn+1to the minimal surfaces.