Aleksandra Borówka

Title: S^6 as nearly Kähler manifold

Abstract: 
It is well known that S^6 admits an almost complex structure
induced by the octonionic multiplication on \mathbb{R}^7=Im\mathbb{O}
which is non-integrable. Together with the standard metric it equips S^6
with an almost-hermitian structure. It turns out that although this
structure is not Kähler (D^gJ=0), it is nearly Kähler, i.e., (D^g_XJ)X.
In this talk I will present basic theory of nearly Kähler manifolds in
dimension 6, including  equivalent definitions using intrinsic torsion
and existence of real Killing spinor. Finally, I will mention very recent
results of L. Foscolo and M. Haskings who showed that S^6 admits an
inhomogenious (in contrary to the standard structure on S^6) nearly
Kähler strucuture - this had been a long lasting open problem.