New $\mathrm{G}_2$-holonomy cones and exotic nearly Kähler structures on $S^6$ and $S^3 \times S^3$

Abstract

Abstract. There is a rich theory of so-called (strict) nearly Kähler manifolds, almostHermitian manifolds generalising the famous almost complex structure on the 6-sphere induced by octonionic multiplication. Nearly Kähler 6-manifolds play a distinguished role both in the general structure theory and also because of their connection with singular spaces with holonomy group the compact exceptional Lie group G2: the metric cone over a Riemannian 6-manifold M has holonomy contained in G2 if and only if M is a nearly Kähler 6-manifold.A central problem in the field has been the absence of any complete inhomogeneous examples. We prove the existence of the first complete inhomogeneous nearly Kähler 6-manifolds by proving the existence of at least one cohomogeneity one nearly Kähler structure on the 6-sphere and on the product of a pair of 3-spheres. We conjecture that these are the only simply connected (inhomogeneous) cohomogeneity one nearly Kähler structures in six dimensions. IntroductionAt least since the early 1950s (see Steenrod's 1951 book [45, 41.22]) it has been well known that viewing S 6 as the unit sphere in Im O endows it with a natural nonintegrable almost complex structure J defined via octonionic multiplication. Since J is compatible with the round metric g rd , the triple (g rd , J, ω), where ω(·, ·) = g rd (J·, ·), defines an almost-Hermitian structure on S 6 . Its torsion has very special properties: in particular, dω is the real part of a complex volume form Ω. Appropriately normalised, the pair (ω, Ω) defines an SU(3)-structure on S 6 which by construction is invariant under the exceptional compact Lie group G 2 Aut(O).Octonionic multiplication also defines a G 2 -invariant 3-form ϕ on Im O byWe call this the standard G 2 -structure on R 7 . Regarding R 7 as the Riemannian cone over (S 6 , g rd ), ϕ and its Hodge dual are given in terms of (ω, Ω):Conversely, viewing S 6 as the level set r = 1 in R 7 , the SU(3)-structure (ω, Ω) is recovered from ϕ and * ϕ by restriction and contraction by the scaling vector field ∂ ∂r . More generally, consider a 7-dimensional Riemannian cone C = C(M ) over a smooth compact manifold (M 6 , g). Suppose that the holonomy of the cone is contained in G 2 . Then there exists a pair of closed (in fact parallel) differential forms ϕ and * ϕ, pointwise equivalent to the model forms on R 7 and homogeneous with respect to scalings on C. Just as above, viewing M as the level set r = 1 in C, the restriction and contraction by ∂ ∂r of ϕ and * ϕ define an SU(3)-structure (ω, Ω) on M satisfying (1.1). In particular, the closedness of ϕ and * ϕ is equivalent to There are other possible equivalent definitions of a nearly Kähler 6-manifold. By relating the holonomy reduction of the cone C(M ) to the existence of a parallel spinor instead of a pair of distinguished parallel forms, nearly Kähler 6-manifolds can be characterised as those 6-manifolds admitting a real Killing spinor [34]. Alternatively, one could give a definition in terms of Gray-Hervella torsion classes of almost Her...

Deformation theory of nearly Kähler manifolds

“…Four of these are homogeneous and were known since 1968 [23]: the round 6-sphere endowed with the non-integrable almost complex structure induced by octonionic multiplication on R 7 Im O and the 3-symmetric spaces CP 3 = Sp(2)/U (1) × Sp (1), S 3 × S 3 = SU (2) 3 /SU (2) and F 3 = SU (3)/T 2 . Recently two inhomogeneous nearly Kähler structures on S 6 and S 3 × S 3 were found in [9]. Finite quotients of the homogeneous nearly Kähler manifolds have also been studied [8].…”

On some hypersurfaces of the homogeneous nearly Kähler

“…B. Butruille showed that the only homogeneous 6-dimensional NK manifolds are the NK 6-sphere 6 , the NK 3 × 3 , the complex projective space 3 and the flag manifold (3)∕ (1) × (1). It is worth mentioning that L. Foscolo and M. Haskins [17] remarkably constructed inhomogeneous NK structures on both manifolds of 6 and 3 × 3 . In order to avoid confusion, from now on of this paper, when we say NK 6 or NK 3 × 3 , we mean always the homogeneous NK 6 or the homogeneous NK 3 × 3 .…”

SU(2)$$^2$$2-invariant $$G_2$$G2-instantons

“…Recall from Proposition 6 that A takes the form in (4.8), determined by functions f ± , g ± : R ≥0 → R. The next result gives the conditions on f ± , g ± so that such A extends over a singular orbit at t = 0. To state it, we observe that Lemma 8 in Appendix A shows that for any SU(2) 2 × U (1)-invariant G 2 -metric which smoothly extends over a singular orbit at t = 0 must be of the form (2.13) for functions A 1 , A 2 = A 3 , B 1 , B 2 = B 3 which admit Taylor expansions of the form 13) for some b ∈ R\{0} and real analytic even functions …”

“…The compatible metric determined by this SU(3) structure on {t} × M is [19] 13) and the resulting metric on R t × M, compatible with the G 2 -structure ϕ = dt ∧ω+ 1 , is given by g = dt 2 + g t . Recall also that this metric has holonomy in G 2 if and only if the SU(3)-structure above solves the Hitchin flow equations (2.2).…”

“…The existence and uniqueness theorem for equations with regular singular points (see [18, chapters 6 and 7], and Theorem 4.7 in [13] for a clearer statement) applies here provided that…”