We introduce and study a notion of invariant intrinsic torsion geometrywhich appears, for instance, in connection with the Bryant-Salamon metric on the spinor bundle over S3. This space is foliated by sixdimensional hypersurfaces, each of which carries a particular type of SO(3)-structure; the intrinsic torsion is invariant under SO(3). The last condition is sufficient to imply local homogeneity of such geometries, and this allows us to give a classification. We close the circle by showing that the Bryant-Salamon metric is the unique complete metric with holonomy G2 that arises from SO(3)-structures with invariant intrinsic torsion.
@article{bwmeta1.element.doi-10_1515_coma-2015-0011,
author = {Diego Conti and Thomas Bruun Madsen},
title = {Invariant torsion and G2-metrics},
journal = {Complex Manifolds},
volume = {2},
year = {2015},
zbl = {1330.53063},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_coma-2015-0011}
}
Diego Conti; Thomas Bruun Madsen. Invariant torsion and G2-metrics. Complex Manifolds, Tome 2 (2015) . http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_coma-2015-0011/
[1] I. Agricola, A. Ferreira, and T. Friedrich. The classification of naturally reductive homogeneous spaces in dimension n 6 6. Differential Geom. Appl., 39:59–92, 2015. | Zbl 06476496
[2] W. Ambrose and I. M. Singer. On homogeneous Riemannian manifolds. Duke Math. J., 25:647–669, 1958. | Zbl 0134.17802
[3] M. Atiyah and E. Witten. M-theory dynamics on a manifold of G2 holonomy. Adv. Theor. Math. Phys., 6(1):1–106, 2002. | Zbl 1033.81065
[4] Y. V. Bazaikin and O. A. Bogoyavlenskaya. Complete Riemannian metrics with holonomy group G2 on deformations of cones over S3 × S3. Math. Notes, 93(5-6):643–653, 2013. Translation of Mat. Zametki 93 (2013), no. 5, 645–657. [WoS][Crossref] | Zbl 1281.53052
[5] L. Bedulli and L. Vezzoni. The Ricci tensor of SU(3)-manifolds. J. Geom. Phys., 57(4):1125–1146, 2007. [Crossref] | Zbl 1126.53017
[6] F. Belgun, V. Cort´es, M. Freibert, and O. Goertsches. On the boundary behaviour of leӔ-invariant Hitchin and hypo flows. J. London Math. Soc., (2015) 92 (1): 41-62. | Zbl 06483984
[7] E. Bonan. Sur des vari´et´es riemanniennes `a groupe d’holonomie G2 ou Spin(7). C. R. Acad. Sci. Paris S´er. A-B, 262:A127– A129, 1966. | Zbl 0134.39402
[8] A. Brandhuber, J. Gomis, S. S. Gubser, and S. Gukov. Gauge theory at large N and new G2 holonomy metrics. Nuclear Phys. B, 611(1-3):179–204, 2001. | Zbl 0971.81071
[9] R. L. Bryant. Calibrated embeddings in the special Lagrangian and coassociative cases. Special issue in memory of Alfred Gray (1939–1998). Ann. Global Anal. Geom. 18(3-4): 405–435, 2000.
[10] R. L. Bryant. Non-embedding and non-extension results in special holonomy. In The many facets of geometry, pages 346– 367. Oxford Univ. Press, Oxford, 2010. | Zbl 1221.53084
[11] R. L. Bryant and S. M. Salamon. On the construction of some complete metrics with exceptional holonomy. Duke Math. J., 58(3):829–850, 1989. | Zbl 0681.53021
[12] S. Chiossi and ´ O. Maci´a. SO(3)-structures on 8-manifolds. Ann. Global Anal. Geom., 43(1):1–18, 2013. | Zbl 1273.53019
[13] S. Chiossi and S. Salamon. The intrinsic torsion of SU(3) and G2 structures. In Differential geometry, Valencia, 2001, pages 115–133. World Sci. Publ., River Edge, NJ, 2002. | Zbl 1024.53018
[14] Z. W. Chong, M. Cvetiˇc, G. W. Gibbons, H. L¨u, C. N. Pope, and P. Wagner. General metrics of G2 holonomy and contraction limits. Nuclear Phys. B, 638(3):459–482, 2002. | Zbl 0997.57047
[15] R. Cleyton and A. Swann. Cohomogeneity-one G2-structures. J. Geom. Phys., 44(2–3):202–220, 2002. [Crossref] | Zbl 1025.53024
[16] R. Cleyton and A. Swann. Einstein metrics via intrinsic or parallel torsion. Math. Z., 247(3): 513–528, 2004. | Zbl 1069.53041
[17] D. Conti and T. B. Madsen. Harmonic structures and intrinsic torsion. Transform. Groups 20(3): 699-723, 2015. | Zbl 1328.53040
[18] M. Cvetiˇc, G. W. Gibbons, H. L¨u, and C. N. Pope. Cohomogeneity one manifolds of Spin(7) and G2 holonomy. Phys. Rev. D (3), 65(10):106004, 29, 2002.
[19] G. W. Gibbons, H. L¨u, C. N. Pope, and K. S. Stelle. Supersymmetric domain walls from metrics of special holonomy. Nuclear Phys. B, 623(1-2):3–46, 2002. | Zbl 1036.83506
[20] D. Giovannini. Special structures and symplectic geometry. PhD thesis, University of Turin, 2003.
[21] N. Hitchin. Stable forms and special metrics. In Global differential geometry: the mathematical legacy of Alfred Gray (Bilbao, 2000), volume 288 of Contemp. Math., pages 70–89. Amer. Math. Soc., Providence, RI, 2001.
[22] S. Karigiannis and J. Lotay. Deformation theory of G2 conifolds. arXiv:1212.6457 [math.DG], 2012.
[23] S. Kobayashi and K. Nomizu. Foundations of differential geometry. Vol I. Interscience Publishers, a division of John Wiley & Sons, New York-Lond on, 1963. | Zbl 0119.37502
[24] Y. G. Nikonorov and E. D. Rodionov. Compact homogeneous Einstein 6-manifolds. Differential Geometry and its Applications, 19(3):369–378, 2003. | Zbl 1045.53031
[25] C. N. Pope. Homogeneous Einstein Metrics on SO(n). arXiv:1001.2776, 2010.
[26] R. Reyes Carri´on. A generalization of the notion of instanton. Differential Geom. Appl., 8(1):1–20, 1998. [Crossref] | Zbl 0902.53036
[27] F. Tricerri and L. Vanhecke. Homogeneous structures on Riemannian manifolds, volume 83 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1983. | Zbl 0509.53043