Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 081, 33 pages      arXiv:1510.03337

A Projective-to-Conformal Fefferman-Type Construction

Matthias Hammerl a, Katja Sagerschnig b, Josef Šilhan c, Arman Taghavi-Chabert d and Vojtĕch Zádník e
a) University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, 1010 Vienna, Austria
b) INdAM-Politecnico di Torino, Dipartimento di Scienze Matematiche, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
c) Masaryk University, Faculty of Science, Kotlářská 2, 61137 Brno, Czech Republic
d) Università di Torino, Dipartimento di Matematica ''G. Peano'', Via Carlo Alberto 10, 10123 Torino, Italy
e) Masaryk University, Faculty of Education, Poříčí 31, 60300 Brno, Czech Republic

Received February 09, 2017, in final form October 09, 2017; Published online October 21, 2017

We study a Fefferman-type construction based on the inclusion of Lie groups ${\rm SL}(n+1)$ into ${\rm Spin}(n+1,n+1)$. The construction associates a split-signature $(n,n)$-conformal spin structure to a projective structure of dimension $n$. 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.

Key words: parabolic geometry; projective structure; conformal structure; Cartan connection; Fefferman spaces; twistor spinors.

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  1. Alt J., On quaternionic contact Fefferman spaces, Differential Geom. Appl. 28 (2010), 376-394, arXiv:1003.1849.
  2. Bailey T.N., Eastwood M.G., Gover A.R., Thomas's structure bundle for conformal, projective and related structures, Rocky Mountain J. Math. 24 (1994), 1191-1217.
  3. Baum H., Friedrich T., Grunewald R., Kath I., Twistor and Killing spinors on Riemannian manifolds, Seminarberichte, Vol. 108, Humboldt Universität, Berlin, 1990.
  4. Calderbank D.M.J., Diemer T., Differential invariants and curved Bernstein-Gelfand-Gelfand sequences, J. Reine Angew. Math. 537 (2001), 67-103, math.DG/0001158.
  5. Čap A., Correspondence spaces and twistor spaces for parabolic geometries, J. Reine Angew. Math. 582 (2005), 143-172, math.DG/0102097.
  6. Čap A., Two constructions with parabolic geometries, Rend. Circ. Mat. Palermo (2) Suppl. (2006), 11-37, math.DG/0504389.
  7. Čap A., Infinitesimal automorphisms and deformations of parabolic geometries, J. Eur. Math. Soc. 10 (2008), 415-437, math.DG/0508535.
  8. Čap A., Gover A.R., CR-tractors and the Fefferman space, Indiana Univ. Math. J. 57 (2008), 2519-2570, math.DG/0611938.
  9. Čap A., Gover A.R., A holonomy characterisation of Fefferman spaces, Ann. Global Anal. Geom. 38 (2010), 399-412, math.DG/0611939.
  10. Čap A., Gover A.R., Hammerl M., Holonomy reductions of Cartan geometries and curved orbit decompositions, Duke Math. J. 163 (2014), 1035-1070, arXiv:1103.4497.
  11. Čap A., Slovák J., Parabolic geometries. I. Background and general theory, Mathematical Surveys and Monographs, Vol. 154, Amer. Math. Soc., Providence, RI, 2009.
  12. Čap A., Slovák J., Souček V., Bernstein-Gelfand-Gelfand sequences, Ann. of Math. 154 (2001), 97-113, math.DG/0001164.
  13. Chevalley C.C., The algebraic theory of spinors, Columbia University Press, New York, 1954.
  14. Crampin M., Saunders D.J., Fefferman-type metrics and the projective geometry of sprays in two dimensions, Math. Proc. Cambridge Philos. Soc. 142 (2007), 509-523.
  15. Dunajski M., Tod P., Four-dimensional metrics conformal to Kähler, Math. Proc. Cambridge Philos. Soc. 148 (2010), 485-503, arXiv:0901.2261.
  16. Eastwood M., Notes on conformal differential geometry, Rend. Circ. Mat. Palermo (2) Suppl. (1996), 57-76.
  17. Eastwood M., Notes on projective differential geometry, in Symmetries and Overdetermined Systems of Partial Differential Equations, IMA Vol. Math. Appl., Vol. 144, Springer, New York, 2008, 41-60, arXiv:0806.3998.
  18. Gover A.R., Laplacian operators and $Q$-curvature on conformally Einstein manifolds, Math. Ann. 336 (2006), 311-334, math.DG/0506037.
  19. Hammerl M., Coupling solutions of BGG-equations in conformal spin geometry, J. Geom. Phys. 62 (2012), 213-223, arXiv:1009.1547.
  20. Hammerl M., Sagerschnig K., Šilhan J., Taghavi-Chabert A., Zádník V., Conformal Patterson-Walker metrics, arXiv:1604.08471.
  21. Hammerl M., Sagerschnig K., Šilhan J., Taghavi-Chabert A., Zádník V., Fefferman-Graham ambient metrics of Patterson-Walker metrics, arXiv:1608.06875.
  22. Hughston L.P., Mason L.J., A generalised Kerr-Robinson theorem, Classical Quantum Gravity 5 (1988), 275-285.
  23. Leitner F., A remark on unitary conformal holonomy, in Symmetries and overdetermined systems of partial differential equations, IMA Vol. Math. Appl., Vol. 144, Springer, New York, 2008, 445-460.
  24. Nurowski P., Projective versus metric structures, J. Geom. Phys. 62 (2012), 657-674, arXiv:1003.1469.
  25. Nurowski P., Sparling G.A., Three-dimensional Cauchy-Riemann structures and second-order ordinary differential equations, Classical Quantum Gravity 20 (2003), 4995-5016, math.DG/0306331.
  26. Patterson E.M., Walker A.G., Riemann extensions, Quart. J. Math. 3 (1952), 19-28.
  27. Penrose R., Rindler W., Spinors and space-time. Vol. 1. Two-spinor calculus and relativistic fields, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1984.
  28. Penrose R., Rindler W., Spinors and space-time. Vol. 2. Spinor and twistor methods in space-time geometry, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1986.
  29. Taghavi-Chabert A., Pure spinors, intrinsic torsion and curvature in even dimensions, Differential Geom. Appl. 46 (2016), 164-203, arXiv:1212.3595.
  30. Taghavi-Chabert A., Twistor geometry of null foliations in complex Euclidean space, SIGMA 13 (2017), 005, 42 pages, arXiv:1505.06938.

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