Locally homogeneous nearly Kähler manifolds

Cortés, Vicente; Vásquez, José J.

doi:10.48550/arxiv.1410.6912

Cited by 2 publications

(3 citation statements)

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“…Furthermore, it is demonstrated in [15] by Foscolo-Haskins that on S 6 and on S 3 × S 3 there is a cohomogeneity one nearly Kähler structure which is not homogeneous. Locally homogeneous examples that are finite quotients of S 3 × S 3 are described in [12] by Cortés-Vásquez. For any t > 0, we have a dilation map t : C → C defined by…”

Section: G 2 Conesmentioning

confidence: 99%

“…This would provide new examples of G 2 cones, and hopefully one could construct new AC G 2 manifolds with these asymptotic cones. In particular, it is worthwhile investigating whether the G 2 cones whose links are the cohomogeneity one nearly Kähler manifolds in [15] or the locally homogeneous nearly Kähler manifolds in [12] actually arise as asymptotic cones of AC G 2 manifolds.…”

Section: Define the Spacementioning

confidence: 99%

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Deformation theory of G_2 conifolds

Karigiannis,

Lotay

2012

Preprint

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We consider the deformation theory of asymptotically conical (AC) and of conically singular (CS) G2 manifolds. In the AC case, we show that if the rate of convergence ν to the cone at infinity is generic in a precise sense and lies in the interval (−4, 0), then the moduli space is smooth and we compute its dimension in terms of topological and analytic data. For generic rates ν < −4 in the AC case, and for generic positive rates of convergence to the cones at the singular points in the CS case, the deformation theory is in general obstructed. We describe the obstruction spaces explicitly in terms of the spectrum of the Laplacian on the link of the cones on the ends, and compute the virtual dimension of the moduli space.We also present many applications of these results, including: the uniqueness of the Bryant-Salamon AC G2 manifolds via local rigidity and the cohomogeneity one property of AC G2 manifolds asymptotic to homogeneous cones; the smoothness of the CS moduli space if the singularities are modeled on particular G2 cones; and the proof of existence of a "good gauge" needed for desingularization of CS G2 manifolds. Finally, we discuss some open problems.

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Section: G 2 Conesmentioning

confidence: 99%

Section: Define the Spacementioning

confidence: 99%

Deformation theory of G_2 conifolds

Karigiannis,

Lotay

2012

Preprint

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“…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, where M = 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.…”

Section: 1mentioning

confidence: 99%

Killing and twistor spinors with torsion

Chrysikos¹

2015

Preprint

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We study twistor spinors (with torsion) on Riemannian spin manifolds (M n , g, T ) carrying metric connections with totally skew-symmetric torsion. We consider the characteristic connection ∇ c = ∇ g + 1 2 T and under the condition ∇ c T = 0, we show that the twistor equation with torsion w.r.t. the family ∇ s = ∇ g + 2sT can be viewed as a parallelism condition under a suitable connection on the bundle Σ ⊕ Σ, where Σ is the associated spinor bundle. Consequently, we prove that a twistor spinor with torsion has isolated zero points. Next we study a special class of twistor spinors with torsion, namely these which are T -eigenspinors and parallel under the characteristic connection; we show that the existence of such a spinor for some s = 1/4 implies that (M n , g, T ) is both Einstein and ∇ c -Einstein, in particular the equation Ric s = Scal s n g holds for any s ∈ R. In fact, for ∇ c -parallel spinors we provide a correspondence between the Killing spinor equation with torsion and the Riemannian Killing spinor equation. This allows us to describe 1-parameter families of non-trivial Killing spinors with torsion on nearly Kähler manifolds and nearly parallel G 2 -manifolds, in dimensions 6 and 7, respectively, but also on the 3-dimensional sphere S 3 . We finally present applications related to the universal and twistorial eigenvalue estimate of the square of the cubic Dirac operator.

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Locally homogeneous nearly Kähler manifolds

Cited by 2 publications

References 0 publications

Deformation theory of G_2 conifolds

Deformation theory of G_2 conifolds

Killing and twistor spinors with torsion

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