Fullness and scalar curvature of the totally real submanifolds in $S⁶(1)$

Abstract

Let M be a totally rea1 3-dimensional submanifold of the nearly Kähler 6-sphere $S⁶(1)$. Theorems are proven on the relation between the fullness and the scalar curvature R of M. In particular, if either R is a constant different from 2, or M is compact with $R≠ 2$, then M is full in $S⁶( 1)$ unless M is totally geodesic. A family of examples with $R≡ 2$, which are fully contained in some great hypersphere $S⁵( 1)⊂ S⁶(1)$, are also defined in an explicit manner.