On almost complex curves in the nearly Kaehler six-sphere

In this paper, we study the curvature properties of almost complex curves M   in the nearly Kaehler six-sphere by using the harmonic sequences theory. For compact almost complex curve of type (I), if the Gaussian curvature K⩽16, then K=16. A basic valued distribution theorem of Gaussian curvature for almost complex curve of type (II) is given. For almost complex curve of type (III), we show that if M   is complete and Gaussian curvature K⩾0K⩾0, then K=0K=0; and if M   is compact and K⩽0K⩽0, then K=0K=0.