Killing spinors on Lorentzian manifolds

The aim of this paper is to describe some results concerning the geometry of Lorentzian manifolds admitting Killing spinors. We prove that there are imaginary Killing spinors on simply connected Lorentzian Einstein–Sasaki manifolds. In the Riemannian case, an odd-dimensional complete simply connected manifold (of dimension n≠7) is Einstein–Sasaki if and only if it admits a non-trivial Killing spinor to . The analogous result does not hold in the Lorentzian case. We give an example of a non-Einstein Lorentzian manifold admitting an imaginary Killing spinor. A Lorentzian manifold admitting a real Killing spinor is at least locally a codimension one warped product with a special warping function. The fiber of the warped product is either a Riemannian manifold with a real or imaginary Killing spinor or with a parallel spinor, or it again is a Lorentzian manifold with a real Killing spinor. Conversely, all warped products of that form admit real Killing spinors.     

Parallel and Killing Spinors on Spinc Manifolds

We describe all simply connected Spin^c manifolds carrying parallel and real Killing spinors. In particular we show that every Sasakian manifold (not necessarily Einstein) carries a canonical Spin^c structure with Killing spinors. Received: 24 December 1996 / Accepted: 6 January 1997     

Twistor spinors on Lorentzian symmetric spaces

An indecomposable Riemannian symmetric space which admits non-trivial twistor spinors has constant sectional curvature. Furthermore, each homogeneous Riemannian manifold with parallel spinors is flat. In the present paper we solve the twistor equation on all indecomposable Lorentzian symmetric spaces explicitly. In particular, we show that there are — in contrast to the Riemannian case — indecomposable Lorentzian symmetric spaces with twistor spinors, which have non-constant sectional curvature and non-flat and non-Ricci flat homogeneous Lorentzian manifolds with parallel spinors.     

Eigenvalue Boundary Problems for the Dirac Operator

 On a compact Riemannian spin manifold with mean-convex boundary, we analyse the ellipticity and the symmetry of four boundary conditions for the fundamental Dirac operator including the (global) APS condition and a Riemannian version of the (local) MIT bag condition. We show that Friedrich's inequality for the eigenvalues of the Dirac operator on closed spin manifolds holds for the corresponding four eigenvalue boundary problems. More precisely, we prove that, for both the APS and the MIT conditions, the equality cannot be achieved, and for the other two conditions, the equality characterizes respectively half-spheres and domains bounded by minimal hypersurfaces in manifolds carrying non-trivial real Killing spinors. Received: 12 November 2001 / Accepted: 25 June 2002 Published online: 21 October 2002 RID="*" ID="*" Research of S. Montiel is partially supported by a Spanish MCyT grant No. BFM2001-2967 and by European Union FEDER funds     

Universal Bose-Fermi mass-relations in Kaluza-Klein supergravity and harmonic analysis on coset manifolds with Killing spinors

Universal mass-relations between Bose and Fermi modes are obtained via general relations existing among the spectra of invariant operators on coset manifolds with Killing spinors. In particular zero-modes for pseudoscalars, spinors, and Lichnerowitz scalars are always related to the isometry and cohomology properties of the manifold. The structure of the four-dimensional supersymmetry rules on physical modes is given. Furthermore, the harmonies of the possible massless multiplets, namely, the gravitational multiplet, the Killing gauge multiplet, and the Betti gauge multiplet, are exhibited for general manifolds. Comments on matter and quasi-massless multiplets conclude the paper.     

Generalised G 2–Manifolds

We define new Riemannian structures on 7–manifolds by a differential form of mixed degree which is the critical point of a (possibly constrained) variational problem over a fixed cohomology class. The unconstrained critical points generalise the notion of a manifold of holonomy G ₂, while the constrained ones give rise to a new geometry without a classical counterpart. We characterise these structures by means of spinors and show the integrability conditions to be equivalent to the supersymmetry equations on spinors in type II supergravity theory with bosonic background fields. In particular, this geometry can be described by two linear metric connections with skew–symmetric torsion. Finally, we construct explicit examples by introducing the device of T–duality.On leave at: Centre de Mathématiques Ecole Polytechnique 91128 Palaiseau, France. E-mail: fwitt@math.polytechnique.fr     

Characterizations of Einstein manifolds and odd-dimensional spheres

In this paper, we use a Killing vector field on a Riemannian manifold to characterize odd-dimensional spheres and Einstein manifolds.     

Twistor forms on Riemannian products

We study twistor forms on products of compact Riemannian manifolds and show that they are defined by Killing forms on the factors. The main result of this note is a necessary step in the classification of compact Riemannian manifolds with non-generic holonomy carrying twistor forms.     

Eigenvalues of the Dirac Operator on Manifolds¶with Boundary

Under standard local boundary conditions or certain global APS boundary conditions, we get lower bounds for the eigenvalues of the Dirac operator on compact spin manifolds with boundary. For the local boundary conditions, limiting cases are characterized by the existence of real Killing spinors and the minimality of the boundary. Received: 22 August 2000 / Accepted: 15 March 2001     

Separable dynamical systems: Characterization of separability structures on Riemannian manifolds

This paper is a direct continuation of the short note [1] on separability structures on Riemannian manifolds. A separability structure on a V_n is characterized by the existence of r Killing vectors and n–r Killing 2-tensors whose properties are briefly collected in a theorem. A general discussion on the form of the metric tensor and the Killing tensors components is given.     

Classification of supersymmetric spacetimes in eleven dimensions

We derive, for spacetimes admitting a Spin(7) structure, the general local bosonic solution of the Killing spinor equation of 11-dimensional supergravity. The metric, four-form, and Killing spinors are determined explicitly, up to an arbitrary eight-manifold of Spin(7) holonomy. It is sufficient to impose the Bianchi identity and one particular component of the four-form field equation to ensure that the solution of the Killing spinor equation also satisfies all the field equations, and we give these conditions explicitly.     

On the significance of Killing tensors

Killing vectors give the linear first integrals of the geodesic equations on Riemannian manifolds and spacetimes, while Killing tensors give the quadratic, cubic, and higher-order first integrals. Here it is shown that the Lie algebra of Killing vectors,, is extended by Killing tensors into a graded algebra,. This sheds some light on the comment by Xanthopoulos [1] on the apparent scarcity of irreducible Killing tensors. Examples are presented of the graded algebras when is abelian and when is nonabelian.     

Supersymmetric Killing Structures

In this text we combine the notions of supergeometry and supersymmetry. We construct a special class of supermanifolds whose reduced manifolds are (pseudo-) Riemannian manifolds. These supermanifolds allow us to treat vector fields on the one hand and spinor fields on the other hand as equivalent geometric objects. This is the starting point of our definition of supersymmetric Killing structures. The latter combines subspaces of vector fields and spinor fields, provided they fulfill certain field equations. This naturally leads to a superalgebra which extends the supersymmetry algebra to the case of non-flat reduced space. We examine in detail the additional terms which enter into this structure and we give a lot of examples.     

Construction of convergent simplicial approximations of quantum fields on Riemannian manifolds

We construct simplicial approximations of random fields on Riemannian manifolds of dimensiond. We prove convergence of the fields to the continuum limit, for arbitraryd in the Gaussian case and ford=2 in the non-Gaussian case. In particular we obtain convergence of the simplicial approximation to the continuum limit for quantum fields on Riemannian manifolds with exponential interaction.Dedicated to Res Jost and Arthur WightmanBiBoS Research Centre     

On nearly parallel G2-structures

A nearly parallel G₂-structure on a seven-dimensional Riemannian manifold is equivalent to a spin structure with a Killing spinor. We prove general results about the automorphism group of such structures and we construct new examples. We classify all nearly parallel G₂-manifolds with large symmetry group and in particular all homogeneous nearly parallel G₂-structures.     

Magnetic fields in 2D and 3D sphere

In this note we study the Landau–Hall problem in the 2D and 3D unit sphere, that is, the motion of a charged particle in the presence of a static magnetic field. The magnetic flow is completely determined for any Riemannian surface of constant Gauss curvature, in particular, the unit 2D sphere. For the 3D case we consider Killing magnetic fields on the unit sphere, and we show that the magnetic flowlines are helices with the given Killing vector field as its axis.     

Invariant solutions to the conformal Killing–Yano equation on Lie groups

We search for invariant solutions of the conformal Killing–Yano equation on Lie groups equipped with left invariant Riemannian metrics, focusing on 2-forms. We show that when the Lie group is compact equipped with a bi-invariant metric or 2-step nilpotent, the only invariant solutions occur on the 3-dimensional sphere or on a Heisenberg group. We classify the 3-dimensional Lie groups with left invariant metrics carrying invariant conformal Killing–Yano 2-forms.     

Eleven-dimensional supergravity and octonions

We analyse an aspect of spontaneous compactification of 11-dimensional simple supergravity on the 7-sphere: the geometric meaning of the pseudo-scalar modes on S₇. In particular, we present the geometrical interpretation of the (anti) self-dual equation and of its 35 solutions. This is deduced from the properties of Killing spinors and of the algebra of octonions.