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Nearly Parallel G2-structures with Large Symmetry Group

Part of: Global differential geometry

Published online by Cambridge University Press:  16 December 2019

Fabio Podestà
Fabio Podestà*
Affiliation:
Dipartimento di Matematica e Informatica “Ulisse Dini”, Università di Firenze, V.le Morgagni 67/A, 50100 Firenze, Italy Email: [email protected]
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Abstract

We prove the existence of a one-parameter family of nearly parallel G2-structures on the manifold $\text{S}^{3}\times \mathbb{R}^{4}$, which are mutually non-isomorphic and invariant under the cohomogeneity one action of the group SU(2)3. This family connects the two locally homogeneous nearly parallel G2-structures that are induced by the homogeneous ones on the sphere S7.

Keywords

Nearly parallel G2-structurescohomogeneity one actionsEinstein metrics

MSC classification

Primary: 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)
Secondary: 53C30: Homogeneous manifolds
Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 2 , April 2021 , pp. 339 - 359
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Copyright
© Canadian Mathematical Society 2019

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References

Albuquerque, R., Variations of gwistor space. Port. Math. 70(2013), 145–160. https://doi.org/10.4171/PM/1929CrossRefGoogle Scholar
Alekseevsky, A. V. and Alekseevsky, D. V., G-manifolds with one dimensional orbit space. In: Lie groups, their discrete subgroups, and invariant theory. Adv. Soviet Math., 8, Amer. Math. Soc., Providence, RI, 1992, pp. 1–31.Google Scholar
Alexandrino, M. and Bettiol, R., Lie Groups and geometric aspects of isometric actions. Springer, Cham, 2015. https://doi.org/10.1007/978-3-319-16613-1CrossRefGoogle Scholar
Alexandrov, B. and Semmelmann, U., Deformations of nearly parallel G 2-structures. Asian J. Math. 16(2012), 713–744. https://doi.org/10.4310/AJM.2012.v16.n4.a6CrossRefGoogle Scholar
Bär, C., Real Killing spinors and holonomy. Comm. Math. Phys. 154(1993), 509–521.CrossRefGoogle Scholar
Bilal, A. and Metzger, S., Compact weak G2-manifolds with conical singularities. Nuclear Phys. B 663(2003), 343–364.CrossRefGoogle Scholar
Boyer, C. P. and Galicki, K., Sasakian geometry, holonomy, and supersymmetry. In: Handbook of Pseudo-Riemannian Geometry and Supersymmetry. IRMA Lectures in Math. and Theor. Physics, 16, Eur. Math. Soc., Zürich, 2010, pp. 39–83. https://doi.org/10.4171/079-1/3CrossRefGoogle Scholar
Böhm, C., Inhomogeneous Einstein metrics on low-dimensional spheres and other low-dimensional spaces. Invent. Math. 134(1998), 145–176. https://doi.org/10.1007/s002220050261Google Scholar
Bryant, R., Metrics with exceptional holonomy. Annals of Math. 126(1987), 525–576. https://doi.org/10.2307/1971360CrossRefGoogle Scholar
Bryant, R. and Harvey, R., Submanifolds in hyper-Kähler geometry. J. Amer. Math. Soc. 2(1989), 1–31. https://doi.org/10.2307/1990911Google Scholar
Bryant, R. and Salamon, S., On the construction of some complete metrics with exceptional holonomy. Duke Math. J. 58(1989), 829–850. https://doi.org/10.1215/S0012-7094-89-05839-0CrossRefGoogle Scholar
Butruille, J.-B., Classification des variétés approximativement kähleriennes homogènes. Ann. Glob. Anal. Geom. 27(2005), 201–225. https://doi.org/10.1007/s10455-005-1581-xCrossRefGoogle Scholar
Cleyton, R. and Swann, A., Cohomogeneity-one G2-structures. J. Geom. Phys. 44(2002), 202–220. https://doi.org/10.1016/S0393-0440(02)00074-8CrossRefGoogle Scholar
Cortés, V., Leistner, T., Schäfer, L., and Schulte-Hegensbach, F., Half-flat structures and special holonomy. J. London Math. Soc. 102(2011), 113–158. https://doi.org/10.1112/plms/pdq012Google Scholar
Fernández, M. and Gray, A., Riemannian manifolds with structure group G2. Ann. Mat. Pura Appl. 32(1982), 19–45. https://doi.org/10.1007/BF01760975CrossRefGoogle Scholar
Fernández, M., Ivanov, S., Muñoz, V., and Ugarte, L., Nearly hypo structures and compact nearly Kähler 6-manifolds with conical singularities. J. Lond. Math. Soc. 78(2008), 580–604. https://doi.org/10.1112/jlms/jdn044CrossRefGoogle Scholar
Foscolo, F. and Haskins, M., New G2-holonomy cones and exotic nearly Kähler structures on S 6 and S 3 × S 3. Ann. of Math. (2) 185(2017), 59–130. https://doi.org/10.4007/annals.2017.185.1.2CrossRefGoogle Scholar
Foscolo, F., Haskins, M., and Nordström, J., Infinitely many new families of complete cohomogeneity one G2-manifolds: G2analogues of the Taub-NUT and Eguchi-Hanson spaces. 2018. arxiv:1805.02612v2Google Scholar
Friedrich, Th., Kath, I., Moroianu, A., and Semmelmann, U., Nearly parallel G2-structures. J. Geom. Phys. 23(1997), 259–286. https://doi.org/10.1016/S0393-0440(97)80004-6CrossRefGoogle Scholar
Gray, A., Weak holonomy groups. Math. Z. 123(1971), 290–300. https://doi.org/10.1007/BF01109983CrossRefGoogle Scholar
Hitchin, N., Stable forms and special metrics. In: Global differential geometry: the mathematical legacy of Alfred Gray (Bilbao, 2000). Contemp. Math., 288, Amer. Math. Soc., Providence, RI, 2001, pp. 70–89. https://doi.org/10.1090/conm/288/04818CrossRefGoogle Scholar
Kobayashi, S., Transformation groups in differential geometry. Springer-Verlag, New York-Heidelberg, 1972.CrossRefGoogle Scholar
Mostow, G. D., The extensibility of local Lie groups of transformations and groups on surfaces. Ann. of Math. 52(1950), 606–636. https://doi.org/10.2307/1969437CrossRefGoogle Scholar
Onishchik, A. L., Topology of transitive transformation groups. Johann Ambrosius Barth Verlag GmbH, Leipzip, 1994.Google Scholar
Podestà, F. and Raffero, A., On the automorphism group of a closed G2-structure. Q. J. Math. 70(2019), 195–200. https://doi.org/10.1093/qmath/hay045Google Scholar
Podestà, F. and Spiro, A., Six-dimensional nearly Kähler manifolds of cohomogeneity one. J. Geom. Phys. 60(2010), 156–164. https://doi.org/10.1016/j.geomphys.2009.09.008CrossRefGoogle Scholar
Podestà, F. and Spiro, A., Six-dimensional nearly Kähler manifolds of cohomogeneity one (II). Comm. Math. Phys. 312(2012), 477–500. https://doi.org/10.1007/s00220-012-1482-3CrossRefGoogle Scholar
Spiro, A., Lie pseudogroups and locally homogeneous Riemannian spaces. Boll. U.M.I. 6B(1992), 843–872.Google Scholar
Stock, S., Lifting SU(3)-structures to nearly parallel G2‐structures. J. Geom. Phys. 59(2009), 1–7. https://doi.org/10.1016/j.geomphys.2008.08.003CrossRefGoogle Scholar
Tanno, S., On the isometry groups of Sasakian manifolds. J. Math. Soc. Japan 22(1970), 579–590. https://doi.org/10.2969/jmsj/02240579CrossRefGoogle Scholar
Ziller, W., Homogeneous Einstein metrics on spheres and projective spaces. Math. Ann. 259(1982), 351–358. https://doi.org/10.1007/BF01456947CrossRefGoogle Scholar