SPECIAL LAGRANGIAN SUBMANIFOLDS OF THE NEARLY KAEHLER 6-SPHERE

Abstract

In this paper, we study Lagrangian submanifolds $M$ of the nearly Kähler 6-sphere $S^6(1)$. It is well known that such submanifolds, which are 3-dimensional, are always minimal and admit a symmetric cubic form. Following an idea of Bryant, developed in the study of Lagrangian submanifolds of $\mathbb C^3$, we then investigate those Lagrangian submanifolds for which at each point the tangent space admits an isometry preserving this cubic form. We obtain that all such Lagrangian submanifolds can be obtained starting from complex curves in $S^6(1)$ or from holomorphic curves in $\mathbb CP^2(4)$. In the final section we classify the Lagrangian submanifolds which admit a Sasakian structure that is compatible with the induced metric. This last result generalizes theorems obtained by Deshmukh and ElHadi. Note that in this case, the condition that $M$ admits a Sasakian structure implies that $M$ admits a pointwise isometry of the tangent space.