Geometric aspects of transversal Killing spinors on Riemannian flows

Ginoux, Nicolas; Habib, Georges

doi:10.1007/s12188-008-0006-8

Cited by 6 publications

(21 citation statements)

References 24 publications

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“…Let (M n+1 , g, F ) be a spin Riemannian flow. Recall [15] that, for α, β ∈ C, an (α, β)-transversal Killing spinor on M is a smooth section ψ ∈ Γ(ΣM) satisfying, for all Z ∈ Γ(Q),…”

Section: Main Theoremmentioning

confidence: 99%

“…the Sonderforschungsbereich 647 "Raum -Zeit -Materie. Analytische und Geometrische Strukturen" of the Deutsche Forschungsgemeinschaft for their support in the preparation of [15] and this paper. It is a pleasure to thank Bernd Ammann, Christian Bär and Oussama Hijazi for valuable comments.…”

Section: Introductionmentioning

confidence: 99%

“…More generally, we study Riemannian flows (see Section 2) which are locally given by Riemannian submersions with 1-dimensional fibres. We are interested in the following question:Can one relate the spectrum of the Dirac operator on a spin manifold submerged onto the space-forms to geometric quantities?The most natural situation to start with consists in assuming the Riemannian flow (M, F ) to carry what we call an (α, β)-transversal Killing spinor and which can be thought of as the lift of some Killing spinor on the base, see [15] for an account of geometrical properties of flows with transversal Killing spinors. Using this spinor field as a test-spinor, we derive eigenvalue estimates for the fundamental Dirac operator of a closed manifold M in terms of the O'Neill tensor [20] of the flow, which is the natural geometric tensor expected in this context.…”

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confidence: 99%

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A spectral estimate for the Dirac operator on Riemannian flows

Ginoux

Habib

2010

Open Mathematics

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We give a new upper bound for the smallest eigenvalues of the Dirac operator on a Riemannian flow carrying transversal Killing spinors. We derive an estimate on Sasakian and on 3-dimensional manifolds and partially classify those satisfying the limiting case. Finally, we compare our estimate with a lower bound in terms of a natural tensor depending on the eigenspinor.Mathematics Subject Classification: 53C12, 53C25, 53C27, 58J50, 35P15For a manifold M n isometrically immersed into one of the three simplyconnected space-forms R n+1 , S n+1 , H n+1 , C. Bär got upper bounds for the eigenvalues of the Dirac operator of M in terms of the mean curvature of the hypersurface [4]. His results follow from the min-max-principle using parallel or Killing spinors as test-spinors.In this paper, we aim at studying the spectrum of the Dirac operator on manifolds arising as total space of submersions over real space forms. More generally, we study Riemannian flows (see Section 2) which are locally given by Riemannian submersions with 1-dimensional fibres. We are interested in the following question:Can one relate the spectrum of the Dirac operator on a spin manifold submerged onto the space-forms to geometric quantities?The most natural situation to start with consists in assuming the Riemannian flow (M, F ) to carry what we call an (α, β)-transversal Killing spinor and which can be thought of as the lift of some Killing spinor on the base, see [15] for an account of geometrical properties of flows with transversal Killing spinors. Using this spinor field as a test-spinor, we derive eigenvalue estimates for the fundamental Dirac operator of a closed manifold M in terms of the O'Neill tensor [20] of the flow, which is the natural geometric tensor expected in this context. We begin with the general framework in Section 3, then we focus our attention on the particular cases where M is Sasakian or 3-dimensional. If M is Sasakian (Section 4), then the general estimate can be substantially simplified and provides the existence of harmonic spinors for suitable deformations of the metric. In Section 5, we give a complete classification of 3-dimensional manifolds satisfying the limiting case: we show that the O'Neill tensor is constant and hence the manifold is either a local Riemannian product or homothetic to a Sasakian manifold.Section 6 sheds a new light on the estimate due to O. Hijazi in terms of the energy-momentum tensor [17]. Indeed we show that, if the limiting case of our estimate is attained on 3-dimensional flows, then so is Hijazi's lower bound (see Proposition 6.2) whereas this fact is still true on Sasakian manifolds for special sections of the spinor bundle of M

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Section: Main Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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A spectral estimate for the Dirac operator on Riemannian flows

Ginoux

Habib

2010

Open Mathematics

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“…From the above calculations we see that∇ ∂y s 1 = 0 and ∇ ∂y s 1 = 0. So, even for the above simple example of Riemannian foliation, we see that the harmonic spinors of the basic Dirac operator are parallel spinors with respect to the above modified connection∇ but not with respect to the classical connection ∇; also, the above spinors are not transverse Killing spinors with the definitions from [18,19,20,21].…”

Section: Geometric Objects Related To the Transverse Geometry Of Riemmentioning

confidence: 90%

Transversal Killing and Twistor Spinors Associated to the Basic Dirac Operators

Ionescu

Slesar

Visinescu³

et al. 2013

Rev. Math. Phys.

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We study the interplay between basic Dirac operator and transversal Killing and twistor spinors. In order to obtain results for general Riemannian foliations with bundle-like metric we consider transversal Killing spinors that appear as natural extension of the harmonic spinors associated with the basic Dirac operator. In the case of foliations with basic-harmonic mean curvature it turns out that this type of spinors coincide with the standard definition. We obtain the corresponding version of classical results on closed Riemannian manifold with spin structure, extending some previous results.

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“…This follows from three facts: every Heisenberg manifold is an S 1 -bundle with totally geodesic fibres over a flat torus; every S 1 -bundle over a manifold carrying parallel spinors carries transversally parallel spinors for the induced spin structure, see e.g. [6,Prop. 3.6]; the whole spinor bundle of any flat torus endowed with its so-called trivial spin structure is trivialized by parallel spinors.…”

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confidence: 99%