Properties of the nearly kähler S3×S3

Abstract

We show how the metric, the almost complex structure and the almost product structure of the homogeneous nearly Kähler S 3 × S 3 can be recovered from a submersion π :

On some hypersurfaces of the homogeneous nearly Kähler

“…Finally, we would recall that after Podestà and Spiro [27], recently Moruz and Vrancken [25] further proved that the following maps…”

“…We would remark that, according to Moruz-Vrancken [25], even though the operator is well-defined it is not necessarily preserved by all the isometries of the NK 3 × 3 . Thus, in this sense, is not really an invariant operator.…”

On Hopf hypersurfaces of the homogeneous nearly Kähler $${\mathbf {S}}^3\times {\mathbf {S}}^3$$

“…Proof Given M, by using the isometry F 2 , we get another Hopf hypersurface F 2 (M) of the NK S 3 × S 3 which also possesses three distinct principal curvatures. From Theorem 5.2 of [22], the differential of the isometry F 2 satisfies the following relationship with J and P: Noticing also that, for any unitary quaternions a, b, c, the isometries F abc and F 2 satisfy (F 2 ) 2 = id and F abc • F 2 = F 2 • F cba . Then, applying for Theorem 5.1 to the hypersurface F 2 (M), we immediately conclude the proof of Theorem 5.3.…”

“…Proof Given M, by using the isometry F 1 , we obviously get another Hopf hypersurface F 1 (M) of the NK S 3 × S 3 which also possesses three distinct principal curvatures. From Theorem 5.1 of [22], we know that the differential of the isometry F 1 anticommutes with the almost complex structure J , and commutes with the almost product structure P, that is,…”

“…As the second main result of this paper, we obtain a classification of them under the additional/natural condition that their holomorphic distributions {U } ⊥ are preserved by the almost product structure P of the homogeneous NK S 3 × S 3 . Before stating the result, we would recall that, according to Moruz and Vrancken [22] and Podestà and Spiro [25], the following three maps (1) F 1 : S 3 × S 3 → S 3 × S 3 with F 1 ( p, q) = (q, p), (2) F 2 : S 3 × S 3 → S 3 × S 3 with F 2 ( p, q) = (p, qp), (3) F abc : S 3 × S 3 → S 3 × S 3 with F abc ( p, q) = (apc, bqc) for any unitary quaternions a, b, c are isometries of the NK S 3 × S 3 . Then, the result can be stated as follows:…”