A survey on Lagrangian submanifolds of nearly Kaehler six-sphere (Q6619031)

As it is known, the Gray-Brown three-fold vector cross-products in the Cayley algebra induce canonical nearly Kähler structures on the six-dimensional sphere \(S^6\). A submanifold \(M\) of the nearly Kähler six-dimensional sphere \(S^6\) is said to be Lagrangian (or totally real) if the operator \(J\) of the almost complex structure maps the tangent bundle of \(M\) onto the normal bundle, and the normal bundle onto the tangent bundle.\N\NThis survey presents the most important results on Lagrangian submanifolds of the nearly Kähler six-sphere obtained by S. Deshmukh, F. Dillen, N. Ejiri, L. Schafer, R. Sharma, K. Smoczyk, L. Vrancken and others. As an example of such a result we mention the following theorem proved by \textit{R. Sharma} [Contemp. Math. 756, 219--228 (2020; Zbl 1459.53031)]:\N\NTheorem. If the almost contact metric structure induced by a unit tangent vector field on a Lagrangian submanifold of the nearly Kählerian six-dimensional sphere is normal, then this almost contact metric structure is Sasakian.\N\NFor the entire collection see [Zbl 1537.53001].