Existence of SU(2)×SU(2) cohomogeneity-one nearly Kähler structures on S2×S4

Determine whether there exists a nearly Kähler structure on the six-manifold S2 × S4 that is invariant under a cohomogeneity-one action of SU(2) × SU(2).

Background

Nearly Kähler geometry in six dimensions is classified via unitary intrinsic torsion and has known homogeneous examples, with recent inhomogeneous cohomogeneity-one structures on S6 and S3×S3 constructed by Foscolo–Haskins. The notes highlight a specific potential case on S2×S4 under SU(2)×SU(2) symmetry where existence is unclear.

Resolving this would expand the catalogue of compact nearly Kähler six-manifolds and test the scope of cohomogeneity-one constructions beyond currently known underlying manifolds.

References

Foscolo and Haskins [55] noted that a nearly Kähler structure with a cohomogeneity one action of SU(2) x SU(2) may exist on S2 x S4, although none is known at present.

— Topics in representation theory and Riemannian geometry (2504.13689 - Russo, 18 Apr 2025) in Remark 6.4, Section 6.2