Locally homogeneous nearly Kähler manifolds

Abstract

We construct locally homogeneous six-dimensional nearly Kähler manifolds as quotients of homogeneous nearly Kähler manifolds M by freely acting finite subgroups of Aut 0 (M). We show that non-trivial such groups do only exists if M = S 3 × S 3 . In that case, we classify all freely acting subgroups of Aut 0 (M) = SU(2) × SU(2) × SU(2) of the form A × B, where A ⊂ SU(2) × SU(2) and B ⊂ SU(2).

Killing and twistor spinors with torsion

“…Together with the standard spheres S 2m , these spaces exhaust all even-dimensional Riemannian manifolds admitting real Killing spinors. Notice that recently in [23], a locally homogeneous nearly Kähler manifold of the form M =M /Γ was described, whereM = S 3 × S 3 and Γ is any finite subgroup of SU 2 × SU 2 . Now, any nearly Kähler manifold admits a characteristic connection ∇ c with parallel skew-torsion, given by T (X, Y ) := (∇ g X J)JY (Gray connection) [31,Thm.…”

Deformations of Nearly Kähler Instantons

“…There are precisely four homogeneous nearly Kähler six-manifolds [11] (see [12] for an English version). Until recently these were the only known complete examples, but within the last year new complete examples have been constructed by taking quotients of the homogeneous examples by freely acting discrete groups [18] and by analysing the ordinary differential equations that describe nearly Kähler metrics with cohomogeneity one [23].…”

The multi-moment maps of the nearly Kähler $$S^3\times S^3$$ S 3 × S 3

“…where i, j, k are imaginary quaternions satisfying i j = k. Note that this frame differs by the one in [4] by a sign in E 3 and F 3 . 3 } be the dual frame. The almost complex structure for the homogenous nearly Kähler S 3 × S 3 is given in this frame by…”

“…The following lemma gives more information aboutν and : 3 . Without loss of generality, by changing coordinates we may assume that C = i.…”

“…The homogeneous spaces S 6 , S 3 × S 3 , CP 3 and SU 3 /T 2 admit strict nearly Kähler structures, and there are no other homogeneous strict nearly Kähler 6-manifolds [2]. There are locally homogeneous structures on quotients of S 3 × S 3 [3], and cohomogeneity one structures on S 6 and S 3 × S 3 , which are conjectured to be the only cohomogeneity one examples [5].…”