arXiv research

I discuss geometry and normal forms for pseudo-Riemannian metrics with parallel spinor fields in some interesting dimensions. I also discuss the interaction of these conditions for parallel spinor fields with the condition that the Ricci tensor vanish (which, for pseudo-Riemannian manifolds, is not an automatic consequ…

The spinor flow stability is proven for Ricci flat metrics and parallel spinor fields.

On a Lorentzian manifold the existence of a parallel null vector field implies certain constraint conditions on the induced Riemannian geometry of a space-like hypersurface. We will derive these constraint conditions and, conversely, show that every real analytic Riemannian manifold satisfying the constraint conditions…

Study infinitesimal deformations of Killing spinors on nearly parallel G2-manifolds.

We provide explicit spinor representations for Clifford algebras.

Study finds holonomy algebras for Lorentzian Weyl spin manifolds with specific spinors.

Classifies manifolds with specific spinors and constructs parallel spinors.

The paper studies energy functionals on spinor bundles and their critical points.

The existence of a recurrent spinor field on a pseudo-Riemannian spin manifold $(M,g)$ is closely related to the existence of a parallel 1-dimensional complex subbundle of the spinor bundle of $(M,g)$. We characterize the following simply connected pseudo-Riemannian manifolds admitting such subbundles in terms of their…

Totally umbilical hypersurfaces in Spin^c manifolds with special spinors have constant mean curvature.

In this paper, we study the existence of a skew Killing spinor (see the definition below) 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…

The paper solves a Cauchy problem for Lorentzian manifolds and classifies manifolds with special holonomy.

The present note deals with the properties of metric connections $\nabla$ with vectorial torsion $V$ on semi-Riemannian manifolds $(M^n,g)$. We show that the $\nabla$-curvature is symmetric if and only if $V^{\flat}$ is closed, and that $V^\perp$ then defines an $(n-1)$-dimensional integrable distribution on $M^n$. If …

The aim of the present paper is to clarify the relationship between immersions of surfaces and solutions of the inhomogeneous Dirac equation. The main idea leading to the description of a surface M^2 by a spinor field is the observation that the restriction to M^2 of any parallel spinor phi on R^3 is (with respect to t…

Study Killing equations for spinor-valued forms on pseudo-Riemannian manifolds.

Survey of connected holonomy groups in Lorentzian manifolds.

Two Spin^c structures characterize hypersurfaces in product spaces.

New formula extends Ricci endomorphism action on spinors, with applications in geometry.

We determine the geometry of supersymmetric heterotic string backgrounds for which all parallel spinors with respect to the connection $\hat\nabla$ with torsion $H$, the NS$\otimes$NS three-form field strength, are Killing. We find that there are two classes of such backgrounds, the null and the timelike. The Killing s…

Investigates parallel spinors on Eguchi-Hanson metrics.

The paper studies parallel spinor flows on 3D Cauchy hypersurfaces and provides initial data characterizations.

We study the full holonomy group of Lorentzian manifolds with a parallel null line bundle. We prove several results that are based on the classification of the restricted holonomy groups of such manifolds and provide a construction method for manifolds with disconnected holonomy which starts from a Riemannian manifold …

New spinor fields reveal local or global geometric properties of manifolds.

Study pp-waves with lightlike parallel spinors in vacuum spacetimes.

Any Spin(7)-manifold admits a metric connection \nabla^c with totally skew-symmetric torsion T^c preserving the underlying structure. We classify those with \nabla^c-parallel T^c\neq0 and non-Abelian isotropy algebra iso(T^c)<spin(7). These are isometric to either Riemannian products or homogeneous naturally reductive …

Parallel spinors help characterize G2* structures and isotropic forms.

It is well-known that 7-dimensional 3-Sasakian manifolds carry a one-parametric family of compatible G_2 structures and that they do not admit a characteristic connection. In this note, we show that there is nevertheless a distinguished cocalibrated G_2 structure in this family whose characteristic connection along wit…

Paper proves a spinor inequality for magnetic fields on spin manifolds.

Spin geometry reviewed with applications in physics.

We classify 7-dimensional cocalibrated $\G_2$-manifolds with parallel characteristic torsion and non-abelian holonomy. All these spaces admit a metric connection $\nabla^{\mathrm{c}}$ with totally skew-symmetric torsion and a spinor field $Ψ_1$ solving the equations in the common sector of type II superstring theory. T…

A four-dimensional Walker geometry is a four-dimensional manifold M with a neutral metric g and a parallel distribution of totally null two-planes. This distribution has a natural characterization as a projective spinor field subject to a certain constraint. Spinors therefore provide a natural tool for studying Walker …

On $Spin^c$ manifolds, we study the Energy-Momentum tensor associated with a spinor field. First, we give a spinorial Gauss type formula for oriented hypersurfaces of a $Spin^c$ manifold. Using the notion of generalized cylinders, we derive the variationnal formula for the Dirac operator under metric deformation and po…

Investigates parallel spinors on Lorentzian four-manifolds using differential geometry.

In this paper we complete the classification of spin manifolds admitting parallel spinors, in terms of the Riemannian holonomy groups. More precisely, we show that on a given n-dimensional Riemannian manifold, spin structures with parallel spinors are in one to one correspondence with lifts to Spin_n of the Riemannian …

Study torsion parallel spinors on Lorentzian 4-manifolds and their evolution flows.

We classify locally homogeneous quasi-Sasakian manifolds in dimension five that admit a parallel spinor $ψ$ of algebraic type $F \cdot ψ= 0$ with respect to the unique connection $\nabla$ preserving the quasi-Sasakian structure and with totally skew-symmetric torsion. We introduce a certain conformal transformation of …

We show that given a conformal structure whose holonomy representation fixes a totally lightlike subspace of arbitrary dimension, there is always a local metric in the conformal class off a singular set which is Ricci-isotropic and gives rise to a parallel, totally lightlike distribution on the tangent bundle. This nat…

Method provides solutions to constraint equations for Lorentzian manifolds with lightlike parallel spinors.

Study on charged parallel spinors and mass-charge inequalities.

New criterion for Ricci-flat manifolds with non-vanishing Rosenberg index.

We describe the possible holonomy groups of simply connected irreducible non-locally symmetric pseudo-Riemannian spin manifolds which admit parallel spinors.

The paper studies semi-Riemannian cones and their geometric properties.

Suppose that $Σ=\partial M$ is the $n$-dimensional boundary of a connected compact Riemannian spin manifold $( M,\langle\;,\;\rangle)$ with non-negative scalar curvature, and that the (inward) mean curvature $H$ of $Σ$ is positive. We show that the first eigenvalue of the Dirac operator of the boundary corresponding to…

Spinor fields depending on tensor fields and other spinor fields are considered. The concept of extended spinor fields is introduced and the theory of differentiation for such fields is developed.

We consider weighted parallel spinors in Lorentzian Weyl geometry in arbitrary dimensions, choosing the weight such that the integrability condition for the existence of such a spinor, implies the geometry to be Einstein-Weyl. We then use techniques developed for the classification of supersymmetric solutions to superg…

Let (M^n,g) be a Riemannian spin manifold. The basic equations in supergravity models of type IIa string theory with 4-form flux involve a 3-form T, a 4-form F, a spinorial covariant derivative \nabla depending on \nabla^g, T, F, and a \nabla-parallel spinor field Ψ. We classify and construct many explicit families of …

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 $\nabla^{c}=\nabla^{g}+\frac{1}{2}T$ and under the condition $\nabla^{c}T=0$, we show that the twistor equation with torsion w.r…

For any triple $(M^n, g, \nabla)$ consisting of a Riemannian manifold and a metric connection with skew-symmetric torsion we introduce an elliptic, second order operator $Ω$ acting on spinor fields. In case of a reductive space and its canonical connection our construction yields the Casimir operator of the isometry gr…