Eigenvalues of the Dirac Operator, Twistors and Killing Spinors on Riemannian Manifolds

Baum, Helga; Friedrich, Thomas

doi:10.1007/978-94-015-8422-7_14

Cited by 112 publications

(327 citation statements)

References 29 publications

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“…In [11], it was proved that the only complete, 6-dimensional, nonKähler nearly Kähler manifolds such that the characteristic connection has reduced holonomy are exactly CP 3 and F (1, 2) (as Riemannian manifolds, both are of course irreducible). For computational details on these very interesting spaces, we refer to [9,Section 5.4]. In fact, one checks that in both cases, the holonomy of ∇ splits the tangent space in three two-dimensional subbundles T 2 i (the upper index indicates the dimension)…”

Section: Examplesmentioning

confidence: 99%

A Note on Generalized Dirac Eigenvalues for Split Holonomy and Torsion

Agricola¹,

Kim²

2014

Bulletin of the Korean Mathematical Society

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Abstract. We study the Dirac spectrum on compact Riemannian spin manifolds M equipped with a metric connection ∇ with skew torsion T ∈ Λ 3 M in the situation where the tangent bundle splits under the holonomy of ∇ and the torsion of ∇ is of 'split' type. We prove an optimal lower bound for the first eigenvalue of the Dirac operator with torsion that generalizes Friedrich's classical Riemannian estimate.

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Section: Examplesmentioning

confidence: 99%

A Note on Generalized Dirac Eigenvalues for Split Holonomy and Torsion

Agricola¹,

Kim²

2014

Bulletin of the Korean Mathematical Society

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“…. 8 its basis (u i in the notation of [2]). One then checks that ψ 3 , ψ 4 , ψ 5 and ψ 6 are fixed under the liftÃd (θ) of the isotropy representation to Spin(7).…”

Section: Torsion Forms With Parallel Spinors On Aloff-wallach Spacesmentioning

confidence: 99%

“…In general, a spinor field ψ of length one defines on a 7-dimensional Riemannian manifold a 3-form of general type by the formula (see [2], [19])…”

Section: Torsion Forms With Parallel Spinors On Aloff-wallach Spacesmentioning

confidence: 99%

On the holonomy of connections with skew-symmetric torsion

Agricola

Friedrich

2004

Mathematische Annalen

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208

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Abstract. We investigate the holonomy group of a linear metric connection with skew-symmetric torsion. In case of the euclidian space and a constant torsion form this group is always semisimple. It does not preserve any non-degenerated 2-form or any spinor. Suitable integral formulas allow us to prove similar properties in case of a compact Riemannian manifold equipped with a metric connection of skew-symmetric torsion. On the Aloff-Wallach space N (1, 1) we construct families of connections admitting parallel spinors. Furthermore, we investigate the geometry of these connections as well as the geometry of the underlying Riemannian metric. Finally, we prove that any 7-dimensional 3-Sasakian manifold admits P 2 -parameter families of linear metric connections and spinorial connections defined by 4-forms with parallel spinors.

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“…These are spinors which fulfill the equation ∇ X φ = −aXφ for a constant a = 0, the Killing number. This equation has been extensively examined in the literature [8,29,21] and in particular [9]. Moreover we would like to stress on [7] where the author draws a remarkable connection between geometric Killing spinors on a manifold and parallel spinors on the cone over the manifold, at least in the Riemannian case.…”

Section: We Have (Dmentioning

confidence: 99%