7-Dimensional compact Riemannian manifolds with Killing spinors

Using a link between Einstein-Sasakian structures and Killing spinors we prove a general construction principle of odd-dimensional Riemannian manifolds with real Killing spinors. In dimensionn=7 we classify all compact Riemannian manifolds with two or three Killing spinors. Finally we classify nonflat 7-dimensional Riemannian manifolds with parallel spinor fields.     

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.     

A simple characterization of generalized Robertson–Walker spacetimes

A generalized Robertson–Walker spacetime is the warped product with base an open interval of the real line endowed with the opposite of its metric and base any Riemannian manifold. The family of generalized Robertson–Walker spacetimes widely extends the one of classical Robertson–Walker spacetimes. Further, generalized Robertson–Walker spacetimes appear as a privileged class of inhomogeneous spacetimes admitting an isotropic radiation. In this section we prove a very simple characterization of generalized Robertson–Walker spacetimes; namely, a Lorentzian manifold is a generalized Robertson–Walker spacetime if and only if it admits a timelike concircular vector field.     

The twistor equation on Riemannian manifolds

It is shown that a twistor spinor on a Riemannian manifold defines a conformal deformation to an Einstein manifold. Twistor spinors on 4-manifolds are considered. A characterization of the hyperbolic space is given. Moreover the solutions of the twistor equation on warped products Mⁿ × , where Mⁿ is an Einstein manifold, are described.     

Green’s Function for the Hodge Laplacian on Some Classes of Riemannian and Lorentzian Symmetric Spaces

We compute the Green’s function for the wave equation on forms on the symmetric spaces M × Σ, where M is a simply connected n-dimensional Riemannian or Lorentzian manifold of constant curvature and Σ is a simply connected Riemannian surface of constant curvature. Our approach is based on a generalization to the case of differential forms of the method of spherical means and on the use of Riesz distributions on manifolds. The radial part of the Green’s function is governed by a fourth order analogue of the Heun equation.     

Optimal Eigenvalues Estimate for the Dirac Operator on Domains with Boundary

We give a lower bound for the eigenvalues of the Dirac operator on a compact domain of a Riemannian spin manifold under the MIT bag boundary condition. The limiting case is characterized by the existence of an imaginary Killing spinor. Mathematics Subject Classifications (2000). Differential Geometry, Global Analysis, 53C27, 53C40, 53C80, 58G25, 83C60.     

Supersymmetry in Lorentzian Curved Spaces and Holography

We consider superconformal and supersymmetric field theories on four-dimensional Lorentzian curved space-times, and their five-dimensional holographic duals. As in the Euclidean signature case, preserved supersymmetry for a superconformal theory is equivalent to the existence of a charged conformal Killing spinor. Differently from the Euclidean case, we show that the existence of such spinors is equivalent to the existence of a null conformal Killing vector. For a supersymmetric field theory with an R-symmetry, this vector field is further restricted to be Killing. We demonstrate how these results agree with the existing classification of supersymmetric solutions of minimal gauged supergravity in five dimensions.     

On supersymmetric Lorentzian Einstein-Weyl spaces

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 supergravity theories to characterise those Lorentzian EW geometries that allow for a weighted parallel spinor, calling the resulting geometries supersymmetric. The overall result is that they are either conformally related to ordinary geometries admitting parallel spinors (w.r.t. the Levi-Cività connection), or they are conformally related to certain Kundt spacetime. A full characterisation is obtained for the 4- and 6-dimensional cases.     

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.     

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     

Classification of the Weyl curvature spinors of neutral metrics in four dimensions

Neutral geometry is of increasing interest. As with Riemannian and Lorentzian geometry, spinors can be expected to provide a valuable tool in neutral geometry. For a neutral metric in four dimensions, the classification of the Weyl curvature spinors by the pattern of principal spinors each admits is given. For each Weyl curvature spinor, there are nine nontrivial types. This classification is then related to the classification, given previously by the author, of a Weyl curvature spinor when regarded as a curvature endomorphism (four types). These results are the neutral analogues of well known and fundamental results in Lorentzian geometry, but display the peculiarities of neutral geometry. One can expect these results to be an essential ingredient in a full understanding of neutral geometry in four dimensions.     

A new characterization of the n-dimensional Einstein static spacetime

The Einstein static spacetime is characterized as the unique geodesically complete and simply connected Lorentzian manifold such that the geodesic flow acts by isometries of the Sasaki metric on any null congruence associated to a conformal timelike vector field.     

Conformal groups and conformally equivalent isometry groups

It is shown that if ann dimensional Riemannian or pseudo-Riemannian manifold admits a proper conformal scalar, every (local) conformal group is conformally isometric, and that if it admits a proper conformal gradient every (local) conformal group is conformally homothetic. In the Riemannian case there is always a conformal scalar unless the metric is conformally Euclidean. In the case of a Lorentzian 4-manifold it is proved that the only metrics with no conformal scalars (and hence the only ones admitting a (local) conformal group not conformally isometric) are either conformal to the plane wave metric with parallel rays or conformally Minkowskian.     

Proof of the Symmetry of the Off-Diagonal¶Hadamard/Seeley–deWitt's Coefficients in¶C ∞ Lorentzian Manifolds by a “Local Wick Rotation”

Completing the results achieved in a previous paper, we prove the symmetry of Hadamard/Seeley-deWitt off-diagonal coefficients in smooth D-dimensional Lorentzian manifolds. This result is relevant because it plays a central rôle in Physics, in particular in the theory of the stress-energy tensor renormalization procedure in quantum field theory in curved spacetime. To this end, it is shown that, in any Lorentzian manifold, a sort of "local Wick rotation" of the metric can be performed provided the metric is a (locally) analytic function of the coordinates and the coordinate are appropriate. No time-like Killing field is necessary. Such a local Wick rotation analytically continues the Lorentzian metric in a neighborhood of any point (more generally, in a neighborhood of a space-like (Cauchy) hypersurface) into a Riemannian metric. The continuation locally preserves geodesically convex neighborhoods. In order to make rigorous the procedure, the concept of a complex pseudo-Riemannian (not Hermitian or Kählerian) manifold is introduced and some features are analyzed. Using these tools, the symmetry of Hadamard/Seeley-deWitt off-diagonal coefficients is proven in Lorentzian analytical manifolds by analytical continuation of the (symmetric) Riemannian heat-kernel coefficients. This continuation is performed in geodesically convex neighborhoods in common with both the metrics. Then, the symmetry is generalized to C^X non analytic Lorentzian manifolds by approximating Lorentzian C^X metrics by analytic metrics in common geodesically convex neighborhoods.     

The geometry of sectional curvatures

A large class of questions in differential geometry involves the relationship between the geometry and the topology of a Riemannian (= positive-definite) manifold. We briefly review the status of the following question from this class: given that a compact, even-dimensional manifold admits a Riemannian metric of positive sectional curvatures, what can one say about its topology? Very few manifolds are known to admit such metrics. For example, is it not known whether or not the product of then-sphere with itself (n ≥ 2) does. One answer to the question above is provided by Synge's theorem: if the manifold is orientable, then it is simply connected. Another possible answer is given by the Hopf conjecture: such a manifold necessarily has positive Euler number. The Hopf conjecture is known to be true for homogeneous manifolds, and for arbitrary manifolds in dimensions two and four. This last result has two, apparently entirely different, proofs, one using Synge's theorem and the other the Gauss-Bonnet formula. Neither, it is shown, can be generalized directly to dimensions six or greater. The Hopf conjecture in these higher dimensions remains open.     

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.     

Some extensions of the Einstein–Dirac equation

We considered an extension of the standard functional for the Einstein–Dirac equation where the Dirac operator is replaced by the square of the Dirac operator and a real parameter controlling the length of spinors is introduced. For one distinguished value of the parameter, the resulting Euler–Lagrange equations provide a new type of Einstein–Dirac coupling. We establish a special method for constructing global smooth solutions of a newly derived Einstein–Dirac system called the CL-Einstein–Dirac equation of type II (see Definition 3.1).