Ambient metrics with exceptional holonomy

Leistner, Thomas; Nurowski, Paweł

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 11 (2012) no. 2, pp. 407-436.

We present conformal structures in signature $(3,2)$ for which the holonomy of the Fefferman-Graham ambient metric is equal to the non-compact exceptional Lie group ${\mathrm{G}}_{2\left(2\right)}$. We write down the resulting 8-parameter family of ${\mathrm{G}}_{2\left(2\right)}$-metrics in dimension seven explicitly in an appropriately chosen coordinate system on the ambient space.

Classification : 53A30, 53B30, 53C29

[1] J. Alt., “Fefferman Constructions in Conformal Holonomy”, PhD thesis, Humboldt-University Berlin, 2008.

[2] S. Armstrong, Definite signature conformal holonomy: a complete classification, J. Geom. Phys. 57 (2007), 2024–2048. | MR

[3] S. Armstrong and T. Leistner, Ambient connections realising conformal tractor holonomy, Monatsh. Math. 152 (2007), 265–282. | MR | Zbl

[4] H. Baum and I. Kath, Parallel spinors and holonomy groups on pseudo-Riemannian spin manifolds, Ann. Global Anal. Geom. 17 (1999), 1–17. | MR | Zbl

[5] M. Berger, Sur les groupes d’holonomie homogène des variétés à connexion affine et des variétés riemanniennes, Bull. Soc. Math. France 83 (1995), 279–330. | EuDML | Numdam | MR | Zbl

[6] R. L. Bryant, Metrics with exceptional holonomy, Ann. of Math. 126 (1987), 525–576. | MR | Zbl

[7] R. L. Bryant, Conformal geometry and 3-plane fields on 6-manifolds, In: “Developments of Cartan Geometry and Related Mathematical Problems”, volume 1502 of RIMS Symposium Proceedings, 2006, 1–15.

[8] R. L. Bryant and S. M. Salamon, On the construction of some complete metrics with exceptional holonomy, Duke Math. J. 58 (1989), 829–850. | MR | Zbl

[9] A. Čap and A. R. Gover, Standard tractors and the conformal ambient metric construction, Ann. Global Anal. Geom. 24 (2003), 231–259. | MR | Zbl

[10] A. Čap and K. Sagerschnig, On Nurowski’s conformal structure associated to a generic rank two distribution in dimension five, J. Geom. Phys. 59 (2009), 901. | MR | Zbl

[11] E. Cartan, Les systèmes de Pfaff, à cinq variables et les équations aux dérivées partielles du second ordre, Ann. Sci. École Norm. Sup. (3) 27 (1910), 109–192. | EuDML | JFM | Numdam | MR

[12] V. Cortés, T. Leistner, L. Schäfer and F. Schulte-Hengesbach, Half-flat structures and special holonomy, Proc. London Math. Soc. 102 (2011), 113–158. | MR | Zbl

[13] B. Doubrov and J. Slovak, Inclusions between parabolic geometries, Pure Appl. Math. Q. 6 (2010), Special Issue: In honor of Joseph J. Kohn, Part 1 of 2, 755–780. | MR | Zbl

[14] C. Fefferman and C. R. Graham, Conformal invariants, Astérisque, (Numero Hors Serie) (1985), The mathematical heritage of Élie Cartan (Lyon, 1984), 95–116. | Numdam | MR | Zbl

[15] C. Fefferman and C. R. Graham, “The Ambient Metric”, Annals of Mathematics Studies, Vol. 178, Princeton University Presse, Princeton, NJ, 2012, x+113 pp. | MR | Zbl

[16] A. R. Gover and F. Leitner, A sub-product construction of Poincaré-Einstein metrics, Internat. J. Math. 20 (2009), 1263–1287. | MR | Zbl

[17] A. R. Gover and P. Nurowski, Obstructions to conformally Einstein metrics in $n$ dimensions, J. Geom. Phys. 56 (2006), 450–484. | MR | Zbl

[18] C. R. Graham and T. Willse, Parallel tractor extension and ambient metrics of holonomy split $G_2$, J. Differential Geom., to appear, arXiv:1109.3504v1 | MR | Zbl

[19] M. Hammerl and K. Sagerschnig, Conformal structures associated to generic rank 2 distributions on 5-manifolds - characterization and Killing-field decomposition, SIGMA Symmetry Integrability Geom. Methods Appl. 5 (2009), Paper 081, 29 pp. | EuDML | MR | Zbl

[20] D. Hilbert, Über den Begriff der Klasse von Differentialgleichungen, Math. Ann. 73 (1912), 95–108. | EuDML | JFM | MR

[21] D. D. Joyce, Compact Riemannian $7$-manifolds with holonomy $G_2$. I, II, J. Differential Geom. 43 (1996), 291–328, 329–375. | MR | Zbl

[22] D. D. Joyce, Compact $8$-manifolds with holonomy $\mathrm{S}pin\left(7\right)$, Invent. Math. 123 (1996), 507–552. | EuDML | MR | Zbl

[23] I. Kath, $G_{2\left(2\right)}^{*}$-structures on pseudo-Riemannian manifolds, J. Geom. Phys. 27 (1998), 155–177. | MR | Zbl

[24] S. Kichenassamy, On a conjecture of Fefferman and Graham, Adv. Math. 184 (2004), 268–288. | MR | Zbl

[25] W. Kopczyński, Pure spinors in odd dimensions, Classical Quantum Gravity 14 (1997), A227–A236. | MR | Zbl

[26] T. Leistner, Conformal holonomy of C-spaces, Ricci-flat, and Lorentzian manifolds, Differential Geom. Appl. 24 (2006), 458–478. | MR | Zbl

[27] T. Leistner and P. Nurowski, Conformal pure radiation with parallel rays, Classical Quantum Gravity 29 (2012), 055007. | MR | Zbl

[28] T. Leistner and P. Nurowski, Ambient metrics for $n$-dimensional $pp$-waves, Comm. Math. Phys. 296 (2010), 881–898. | MR | Zbl

[29] F. Leitner, Normal conformal Killing forms, 2004. arXiv.org:math/0406316. | MR

[30] F. Leitner, Conformal Killing forms with normalisation condition, Rend. Circ. Mat. Palermo (2) Suppl., 75 (2005), 279–292. | MR | Zbl

[31] P. Nurowski, Differential equations and conformal structures, J. Geom. Phys. 43 (2005), 327–340. | MR | Zbl

[32] P. Nurowski, Conformal structures with explicit ambient metrics and conformal $G_2$ holonomy, In: “Symmetries and Overdetermined Systems of Partial Differential Equations”, Vol. 144 of IMA Vol. Math. Appl., Springer, New York, 2008, 515–526. | MR | Zbl

[33] S. M. Salamon, “Riemannian Geometry and Holonomy Groups”, Vol. 201 of Pitmann Research Lecture Notes, 1989. | MR | Zbl

[34] H. Wu, On the de Rham decomposition theorem, Illinois J. Math. 8 (1964), 291–311. | MR | Zbl