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...

We develop a powerful new analytic method to construct complete non-compact Ricciflat 7-manifolds, more specifically G2-manifolds, i.e. Riemannian 7-manifolds (M, g) whose holonomy group is the compact exceptional Lie group G2. Our construction gives the first general analytic construction of complete non-compact Ricci-flat metrics in any odd dimension and establishes a link with the Cheeger-Fukaya-Gromov theory of collapse with bounded curvature.The construction starts with a complete non-compact asymptotically conical Calabi-Yau 3-fold B and a circle bundle M → B satisfying a necessary topological condition. Our method then produces a 1-parameter family of circle-invariant complete G2-metrics gǫ on M that collapses with bounded curvature as ǫ → 0 to the original Calabi-Yau metric on the base B. The G2-metrics we construct have controlled asymptotic geometry at infinity, so-called asymptotically locally conical (ALC) metrics; these are the natural higher-dimensional analogues of the ALF metrics that are well known in 4-dimensional hyperkähler geometry.We give two illustrations of the strength of our method. Firstly we use it to construct infinitely many diffeomorphism types of complete non-compact simply connected G2-manifolds; previously only a handful of such diffeomorphism types was known. Secondly we use it to prove the existence of continuous families of complete non-compact G2-metrics of arbitrarily high dimension; previously only rigid or 1-parameter families of complete non-compact G2-metrics were known.

Nearly Kähler manifolds are the Riemannian 6-manifolds admitting real Killing spinors. Equivalently, the Riemannian cone over a nearly Kähler manifold has holonomy contained in G2. In this paper we study the deformation theory of nearly Kähler manifolds, showing that it is obstructed in general. More precisely, we show that the infinitesimal deformations of the homogeneous nearly Kähler structure on the flag manifold are all obstructed to second order.

We construct infinitely many new 1-parameter families of simply connected complete non-compact G 2 -manifolds with controlled geometry at infinity. The generic member of each family has so-called asymptotically locally conical (ALC) geometry. However, the nature of the asymptotic geometry changes at two special parameter values: at one special value we obtain a unique member of each family with asymptotically conical (AC) geometry; on approach to the other special parameter value the family of metrics collapses to an AC Calabi-Yau 3-fold. Our infinitely many new diffeomorphism types of AC G 2 -manifolds are particularly noteworthy: previously the three examples constructed by Bryant and Salamon in 1989 furnished the only known simply connected AC G 2 -manifolds.We also construct a closely related conically singular G 2 -holonomy space: away from a single isolated conical singularity, where the geometry becomes asymptotic to the G 2 -cone over the standard nearly Kähler structure on the product of a pair of 3-spheres, the metric is smooth and it has ALC geometry at infinity. We argue that this conically singular ALC G 2 -space is the natural G 2 analogue of the Taub-NUT metric in 4-dimensional hyperKähler geometry and that our new AC G 2 -metrics are all analogues of the Eguchi-Hanson metric, the simplest ALE hyperKähler manifold. Like the Taub-NUT and Eguchi-Hanson metrics, all our examples are cohomogeneity one, i.e. they admit an isometric Lie group action whose generic orbit has codimension one.

We construct infinitely many new 1-parameter families of simply connected complete noncompact G2-manifolds with controlled geometry at infinity. The generic member of each family has so-called asymptotically locally conical (ALC) geometry. However, the nature of the asymptotic geometry changes at two special parameter values: at one special value we obtain a unique member of each family with asymptotically conical (AC) geometry; on approach to the other special parameter value the family of metrics collapses to an AC Calabi-Yau 3-fold. Our infinitely many new diffeomorphism types of AC G2-manifolds are particularly noteworthy: previously the three examples constructed by Bryant and Salamon in 1989 furnished the only known simply connected AC G2-manifolds.We also construct a closely related conically singular G2-holonomy space: away from a single isolated conical singularity, where the geometry becomes asymptotic to the G2-cone over the standard nearly Kähler structure on the product of a pair of 3-spheres, the metric is smooth and it has ALC geometry at infinity. We argue that this conically singular ALC G2-space is the natural G2 analogue of the Taub-NUT metric in 4-dimensional hyperKähler geometry and that our new AC G2-metrics are all analogues of the Eguchi-Hanson metric, the simplest ALE hyperKähler manifold. Like the Taub-NUT and Eguchi-Hanson metrics, all our examples are cohomogeneity one, i.e. they admit an isometric Lie group action whose generic orbit has codimension one.