Parallel spinors on globally hyperbolic Lorentzian four-manifolds

Abstract

We investigate the differential geometry and topology of globally hyperbolic fourmanifolds (M, g) admitting a parallel real spinor ε. Using the theory of parabolic pairs recently introduced in [22], we first formulate the parallelicity condition of ε on M as a system of partial differential equations, the parallel spinor flow equations, for a family of polyforms on any given Cauchy surface Σ ֒→ M . Existence of a parallel spinor on (M, g) induces a system of constraint partial differential equations on Σ, which we prove to be equivalent to an exterior differential system involving a cohomological condition on the shape operator of the embedding Σ ֒→ M . Solutions of this differential system are precisely the allowed initial data for the evolution problem of a parallel spinor and define the notion of parallel Cauchy pair (e, Θ), where e is a coframe and Θ is a symmetric two-tensor. We characterize all parallel Cauchy pairs on simply connected Cauchy surfaces, refining a result of Baum, Leistner and Lischewski. Furthermore, we classify all compact three-manifolds admitting parallel Cauchy pairs, proving that they are canonically equipped with a locally free action of R 2 and are isomorphic to certain torus bundles over S 1 , whose Riemannian structure we characterize in detail. Moreover, we classify all left-invariant parallel Cauchy pairs on simply connected Lie groups, specifying when they are allowed initial data for the Ricci flat equations and when the shape operator is Codazzi. Finally, we give a novel geometric interpretation of a class of parallel spinor flows and solve it in several examples, obtaining explicit families of four-dimensional Lorentzian manifolds carrying parallel spinors.

Parallel spinor flows on three-dimensional Cauchy hypersurfaces

“…These tuples contain the information about both the real parallel spinor and the underlying globally hyperbolic Lorentzian metric. The following theorem generalizes [23,Theorem 5.4] to the case in which {β t } t∈I is not necessarily constant.…”

“…In this section we briefly review the theory of parallel spinors on globally hyperbolic Lorentzian four-dimensional manifolds as developed in [14,23], where parallel spinors were characterized in terms of a certain type of distribution satisfying a prescribed system of partial differential equations.…”

“…Using this formalism, our first result (see Theorem 3.2) reformulates the evolution problem of a parallel spinor as a system of flow equations for a family of functions {β t } t∈I and a family of coframes {e t } t∈I on an appropriately chosen Cauchy hypersurface Σ ⊂ M . This system of equations yields a generalization of [23,Theorem 5.4] to the case in which the family {β t } t∈I is non-trivial and defines the notion of parallel spinor flow as a family {β t , e t } t∈I satisfying equations (3.6) and (3.7). Using the notion of parallel spinor flow, the constraint equations of the initial value problem of a parallel spinor can be shown to be equivalent to a differential system, the parallel Cauchy differential system [23], for a pair (e, Θ), where e is a coframe and Θ is a symmetric two-tensor on Σ.…”

“…This system of equations yields a generalization of [23,Theorem 5.4] to the case in which the family {β t } t∈I is non-trivial and defines the notion of parallel spinor flow as a family {β t , e t } t∈I satisfying equations (3.6) and (3.7). Using the notion of parallel spinor flow, the constraint equations of the initial value problem of a parallel spinor can be shown to be equivalent to a differential system, the parallel Cauchy differential system [23], for a pair (e, Θ), where e is a coframe and Θ is a symmetric two-tensor on Σ. One of the salient features of this formulation is that the Riemannian h metric induced by (M, g) on Σ is given by the canonical metric for which e = (e u , e l , e n ) becomes an orthonormal coframe, that is, h e = e u ⊗ e u + e l ⊗ e l + e n ⊗ e n .…”

“…Another important feature of parallel spinorial flows is that they admit a canonical notion of leftinvariance in terms of which we can define the notion of left-invariance of parallel spinors. The classification of all left-invariant parallel Cauchy pairs on simply connected three-dimensional Lie groups was completed in [23]. We use this result to obtain the classification of all associated leftinvariant parallel spinor flows.…”