36 research outputs found
Pure spinors, intrinsic torsion and curvature in odd dimensions
We study the geometric properties of a -dimensional complex manifold
admitting a holomorphic reduction of the frame bundle to the
structure group , the stabiliser of
the line spanned by a pure spinor at a point. Geometrically, is
endowed with a holomorphic metric , a holomorphic volume form, a spin
structure compatible with , and a holomorphic pure spinor field up to
scale. The defining property of is that it determines an almost null
structure, i.e.\ an -plane distribution along which is
totally degenerate.
We develop a spinor calculus, by means of which we encode the geometric
properties of and of its rank- orthogonal complement
corresponding to the algebraic properties of the
intrinsic torsion of the -structure. This is the failure of the Levi-Civita
connection of to be compatible with the -structure. In a
similar way, we examine the algebraic properties of the curvature of .
Applications to spinorial differential equations are given. Notably, we
relate the integrability properties of and
to the existence of solutions of odd-dimensional
versions of the zero-rest-mass field equation. We give necessary and sufficient
conditions for the almost null structure associated to a pure conformal Killing
spinor to be integrable. Finally, we conjecture a Goldberg--Sachs-type theorem
on the existence of a certain class of almost null structures when
has prescribed curvature.
We discuss applications of this work to the study of real pseudo-Riemannian
manifolds.Comment: Odd-dimensional version of arXiv:1212.3595 v2: Presentation improved.
A number of corrections made: diagrams describing the curvature and intrinsic
torsion classification; Geometric interpretation of spinorial equations; some
errors in formulae now fixed. Some material regarding parallel spinors
removed (to be including in a separate article) v3: as publishe
Twistor Geometry of Null Foliations in Complex Euclidean Space
We give a detailed account of the geometric correspondence between a smooth
complex projective quadric hypersurface of dimension , and its twistor space , defined to be the space of all linear
subspaces of maximal dimension of . Viewing complex Euclidean
space as a dense open subset of , we show how
local foliations tangent to certain integrable holomorphic totally null
distributions of maximal rank on can be constructed in terms of
complex submanifolds of . The construction is illustrated by means
of two examples, one involving conformal Killing spinors, the other, conformal
Killing-Yano -forms. We focus on the odd-dimensional case, and we treat the
even-dimensional case only tangentially for comparison
Pure spinors, intrinsic torsion and curvature in even dimensions
We study the geometric properties of a -dimensional complex manifold
admitting a holomorphic reduction of the frame bundle to the
structure group , the stabiliser of the
line spanned by a pure spinor at a point. Geometrically, is
endowed with a holomorphic metric , a holomorphic volume form, a spin
structure compatible with , and a holomorphic pure spinor field up to
scale. The defining property of is that it determines an almost null
structure, ie an -plane distribution along which is
totally degenerate.
We develop a spinor calculus, by means of which we encode the geometric
properties of corresponding to the algebraic properties of
the intrinsic torsion of the -structure. This is the failure of the
Levi-Civita connection of to be compatible with the -structure.
In a similar way, we examine the algebraic properties of the curvature of
.
Applications to spinorial differential equations are given. In particular, we
give necessary and sufficient conditions for the almost null structure
associated to a pure conformal Killing spinor to be integrable. We also
conjecture a Goldberg-Sachs-type theorem on the existence of a certain class of
almost null structures when has prescribed curvature.
We discuss applications of this work to the study of real pseudo-Riemannian
manifolds.Comment: v2. Cleaned up version. Typos and errors fixed. Some reordering. v3.
Restructured - some material moved to an additional appendix for clarity -
further typos fixed and other minor improvements v4. Presentation improved.
Some material removed to be included in a future article. v5. As published:
Abstract and intro rewritten. Presentation simplifie
A Goldberg-Sachs theorem in dimension three
We prove a Goldberg-Sachs theorem in dimension three. To be precise, given a
three-dimensional Lorentzian manifold satisfying the topological massive
gravity equations, we provide necessary and sufficient conditions on the
tracefree Ricci tensor for the existence of a null line distribution whose
orthogonal complement is integrable and totally geodetic. This includes, in
particular, Kundt spacetimes that are solutions of the topological massive
gravity equations.Comment: 31 pages. v2: minor typographic changes in the bibliograph
Optical structures, algebraically special spacetimes, and the Goldberg-Sachs theorem in five dimensions
Optical (or Robinson) structures are one generalisation of four-dimensional
shearfree congruences of null geodesics to higher dimensions. They are
Lorentzian analogues of complex and CR structures. In this context, we extend
the Goldberg-Sachs theorem to five dimensions. To be precise, we find a new
algebraic condition on the Weyl tensor, which generalises the Petrov type II
condition, in the sense that it ensures the existence of such congruences on a
five-dimensional spacetime, vacuum or under weaker assumptions on the Ricci
tensor. This results in a significant simplification of the field equations. We
discuss possible degenerate cases, including a five-dimensional generalisation
of the Petrov type D condition. We also show that the vacuum black ring
solution is endowed with optical structures, yet fails to be algebraically
special with respect to them. We finally explain the generalisation of these
ideas to higher dimensions, which has been checked in six and seven dimensions.Comment: Final corrected version (same as published). Remark 2.9 corrected.
Corollary 3.10 clarifie
A Projective-to-Conformal Fefferman-Type Construction
We study a Fefferman-type construction based on the inclusion of Lie groups
into . The construction associates a
split-signature -conformal spin structure to a projective structure of
dimension . We prove the existence of a canonical pure twistor spinor and a
light-like conformal Killing field on the constructed conformal space. We
obtain a complete characterisation of the constructed conformal spaces in terms
of these solutions to overdetermined equations and an integrability condition
on the Weyl curvature. The Fefferman-type construction presented here can be
understood as an alternative approach to study a conformal version of classical
Patterson-Walker metrics as discussed in recent works by Dunajski-Tod and by
the authors. The present work therefore gives a complete exposition of
conformal Patterson-Walker metrics from the viewpoint of parabolic geometry
Optical geometries
We study the notion of optical geometry, defined to be a Lorentzian manifold
equipped with a null line distribution, from the perspective of intrinsic
torsion. This is an instance of a non-integrable version of holonomy reduction
in Lorentzian geometry. These generate congruences of null curves, which play
an important r\^{o}le in general relativity. Conformal properties of these are
investigated. We also extend this concept to generalised optical geometries as
introduced by Robinson and Trautman.Comment: 46 pages; v2: typos fixed, some clarifications added, references
added; v3: references added, minor structural and explanatory change