Skew Killing spinors

We study the existence of a skew Killing spinor on 2- and 3-dimensional Riemannian spin manifolds. We establish the integrability conditions and prove that these spinor fields correspond to twistor spinors in the two dimensional case while, up to a conformal change of the metric, they correspond to parallel spinors in the three dimensional case.     

Generalized Killing spinors on spheres

We study generalized Killing spinors on round spheres \(\mathbb {S}^n\) . We show that on the standard sphere \(\mathbb {S}^8\) any generalized Killing spinor has to be an ordinary Killing spinor. Moreover, we classify generalized Killing spinors on \(\mathbb {S}^n\) whose associated symmetric endomorphism has at most two eigenvalues and recover in particular Agricola–Friedrich’s canonical spinor on 3-Sasakian manifolds of dimension 7. Finally, we show that it is not possible to deform Killing spinors on standard spheres into genuine generalized Killing spinors.     

Skew Killing spinors in four dimensions

Annals of Global Analysis and Geometry - This paper is devoted to the classification of 4-dimensional Riemannian spin manifolds carrying skew Killing spinors. A skew Killing spinor $$psi $$ is a...     

On spinors

For a 2^n-dimensional complex Hermitian vector space S, we prove that any unitary basis of S can be explained as an augmented spinor structure on S. By using this explanation, a SpinC（2n）- action on S is equivalent to an action on a subset of augmented spinor structures. The latter action is a little easy to be understood, and is shown in the last part of this paper. Such kind of understanding could be of use to the discussions of Hermitian manifolds and spin manifolds, especially could help to find connections and elliptical operators.     

Symmetric spinors ink-dimensions

Summary In his paper, explicit formulae are given for any irreducible spinor, in any number of dimensions, which is symmetric in its suffixes. The method employed in determining these formulae is immediately applicable to the task of obtaining explicit forms for spinors which are unsymmetric in their suffixes. Entrata in Redazione il 22 maggio 1971.     

Almost harmonic spinors

We show that any closed spin manifold not diffeomorphic to the two-sphere admits a sequence of volume-one-Riemannian metrics for which the smallest non-zero Dirac eigenvalue tends to zero. As an application, we compare the Dirac spectrum with the conformal volume.     

Metrics with harmonic spinors

We show that every closed spin manifold of dimensionn 3 mod 4 with a fixed spin structure can be given a Riemannian metric with harmonic spinors, i.e. the corresponding Dirac operator has a non-trivial kernel (Theorem A). To prove this we first compute the Dirac spectrum of the Berger spheresS ⁿ ,n odd (Theorem 3.1). The second main ingredient is Theorem B which states that the Dirac spectrum of a connected sumM ₁#M ₂ with certain metrics is close to the union of the spectra ofM ₁ and ofM ₂.Partially supported by SFB 256 and by the GADGET program of the EU     

Surgery and harmonic spinors

Let M be a compact spin manifold with a chosen spin structure. The Atiyah-Singer index theorem implies that for any Riemannian metric on M the dimension of the kernel of the Dirac operator is bounded from below by a topological quantity depending only on M and the spin structure. We show that for generic metrics on M this bound is attained.     

Harmonic spinors on Riemann surfaces

We calculate the dimension of the space of harmonic spinors on hyperelliptic Riemann surfaces for all spin structures. Furthermore, we present non-hype relliptic examples of genus 4 and 6 on which the maximal possible number of linearly independent harmonic spinors is achieved.The second author was supported by the Schweizerischer Nationalfonds zur Förderung wissenschaftlicher Forschung     

Parallel pure spinors and holonomy

We characterize the spin pseudo-Riemannian manifolds which admit parallel pure spinors by their holonomy groups. In particular, we study the Lorentzian case. To cite this article: A. Ikemakhen, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Harmonic symplectic spinors on Riemann surfaces

Symplectic spinor fields were considered already in the 70th in order to give the construction of half-densities in the context of geometric quantization. We introduced symplectic Dirac operators acting on symplectic spinor fields and started a systematical investigation. In this paper, we motivate the notion of harmonic symplectic spinor fields. We describe how many linearly independent harmonic symplectic spinors each Riemann surface admits. Furthermore, we calculate the spectrum of the symplectic spinor Laplacian on the complex projective space of complex dimension 1.     

Describing spinors using probability distribution functions

Irreducible representations of the rotation group are realized using a family of positive probability distributions of the spin projections for an arbitrary value of the spin. The family is parametrized by the points on the sphere. An invertible mapping of the spinors onto the probability distribution functions is constructed. Examples of probability distributions for the well-known states with the spins 1/2 and 1 are presented. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 115, No. 2, pp. 185–198. May, 1998.     

The Dirac operator on symplectic spinors

Symplectic spinors were introduced by B. Kostant in [4] in the context of geometric quantization. This paper presents further considerations concerning symplectic spinors. We define the spinor derivative induced by a symplectic covariant derivative. We compute an explicit formula for this spinor derivative and prove some elementary properties. This makes it possible to define the symplectic Dirac operator in a canonical way. In case of a symplectic and torsion-free covariant derivative it turns out to be formally selfadjoint.