Calabi Lifting and Surface Geometry in S⁴

M$ is an orientable surface in the 4-dimensional Euclidean space $R⁴$ , we can define the Gauss map of $M$ into $S²× S²$ .We note that the Gauss map is constructed as follows: By the same argument as above, each point of $M$ gives an orthogonal complex structure of $R⁴$ compatible with the fixed orientation of $R⁴$ .Since the space of orthogonal complex structures of $R⁴$ is $S²$ , we obtain a map of $M$ into $S²$ .Changing the orientation of $R⁴$ , we get another map of $M$ into $S²$ .It is easy to see that these maps of $M$ into $S²$ give the Gauss map of $M$ into $S²× S²$ .From this point of view, we may regard Calabi lifting as "Gauss map".In this paper, we investigate some relations between an isometric immersion into $S⁴(1)$ and its Calabi lifting.In the sections 1, 2 and 3, we review the results of Chern [7] and Barbosa [2] on minimal surfaces