Totally real submanifolds in $S^6(1)$ satisfying Chen’s equality

Abstract

In this paper, we study 3-dimensional totally real submanifolds of $S^{6}(1)$. If this submanifold is contained in some 5-dimensional totally geodesic $S^{5}(1)$, then we classify such submanifolds in terms of complex curves in $\mathbb {C}P^{2}(4)$ lifted via the Hopf fibration $S^{5}(1)\to \mathbb {C}P^{2}(4)$. We also show that such submanifolds always satisfy Chen’s equality, i.e. $\delta _{M} = 2$, where $\delta _{M}(p)=\tau (p)-\inf K(p)$ for every $p\in M$. Then we consider 3-dimensional totally real submanifolds which are linearly full in $S^{6}(1)$ and which satisfy Chen’s equality. We classify such submanifolds as tubes of radius $\pi /2$ in the direction of the second normal space over an almost complex curve in $S^{6}(1)$.