Calibrations and Lagrangian submanifolds in the six sphere
It is well-known that \(S^6\) carries a natural non-integrable almost complex structure with respect to which the standard metric is Hermitian (but not Kähler). The natural 2-form is non-degenerate but not closed. Unlike in the integrable Kähler case, \textit{N. Ejiri} [Proc. Am. Math. Soc. 83, 759-763 (1981; Zbl 0474.53051)] proved that any smooth Lagrangian submanifold in \(S^6\) is minimal with respect to the standard metric. Based on Ejiri's result, the paper under review studies variational properties of Lagrangian submanifolds and derives the second variational formula for them. The author then applies the formula to the case of certain natural Lagrangian submanifolds in \(S^6\) that are obtained from complex curves in \({\mathbb C} P^2\) through the Hopf fibration. The paper also describes the calibration property of the Lagrangian submanifolds.
scientific article; zbMATH DE number 1230538
scientific article