On the geometry of closed \(G_{2}\)-structures. (English) Zbl 1122.53026

Let \(e_1,\dots, e_7\) be an oriented orthonormal basis of \(\mathbb R^7\) and denote \(e_{i_1}\wedge e_{i_2}\wedge\cdots\wedge e_{i_p}\) by \(e_{i_i}\cdots e_{i_p}\). The subgroup of \(\text{GL}(7)\) that fixes the three-form
\[ \omega= e_{124}+ e_{235}+ e_{346}+ e_{457}+ e_{561}+ e_{672}+ e_{713} \]
is the exceptional Lie group \(G_2\) and is a compact, connected, simply-connected, simple Lie subgroup of \(\text{SO}(7)\) of \(\dim 14\). A \(G_2\)-structure on a 7-manifold is a reduction of the structure group of the tangent bundle to \(G_2\), and \(\omega\) is called the fundamental form of the \(G_2\)-manifold \(M\). When \(\omega\) is parallel, it is known that the holonomy group is contained in \(G_2\). Compact examples of such parallel \(G_2\)-manifolds were obtained by D. Joyce [J. Differ. Geom 43, No. 2, 291–328 (1996; Zbl 0861.53022)] which are all called Joyce spaces.
In this paper the authors study the geometry of closed \(G_2\)-structures and present as main theorem the following result: A compact and closed \(G_2\)-manifold with harmonic Weyl tensor is a Joyce space. With this theorem associated is the following corollary: If \(M\) as above admits a harmonic curvature or is an Einstein space, then it is a Joyce space.


53C29 Issues of holonomy in differential geometry
53C10 \(G\)-structures


Zbl 0861.53022
Full Text: DOI arXiv


[1] Bonan E. (1966) Sur le variétés riemanniennes a groupe d’holonomie G 2 ou Spin(7). C. R. Acad. Sci. Paris 262, 127–129 · Zbl 0134.39402
[2] Bryant R. (1987) Metrics with exceptional holonomy. Ann. Math. 126, 525–576 · Zbl 0637.53042
[3] Bryant, R.: Some remarks on G 2-structures. http://arxiv.org/list/math.DG/0305124, 2003
[4] Bryant, R.: Some remarks on G 2-structures. To appear in the 2005 Gkova Geometry/Topology Conference Proceedings, Cambridge, MA: International Press · JFM 17.0045.03
[5] Bryant R., Salamon S. (1989) On the construction of some complete metrics with exceptional holonomy. Duke Math. J. 58, 829–850 · Zbl 0681.53021
[6] Cabrera F., Monar M., Swann A. (1996) Classification of G 2-structures. J. London Math. Soc. 53, 407–416 · Zbl 0861.53024
[7] Chiossi, S., Salamon, S.: The intrinsic torsion of SU(3) and G 2-structures. In: Differential Geometry, Valencia 2001, River Edge, NJ: World Sci. Publishing, 2002, pp. 115–133 · Zbl 1024.53018
[8] Cleyton, R., Ivanov, S.: On the geometry of closed G 2- structures. http://arxiv.org/list/math.DG/0306362, 2003 · Zbl 1122.53026
[9] Cvetič M., Gibbons G.W., Lu H., Pope C.N.(2002) Almost Special Holonomy in Type IIA&M Theory. Nucl. Phys. B638:186–206 · Zbl 0997.83074
[10] Fernández M. (1987) An Example of compact calibrated manifold associated with the exceptional Lie group G 2. J. Diff. Geom. 26, 367–370 · Zbl 0604.53013
[11] Fernández, M., Gray, A.: Riemannian manifolds with structure group G 2. Ann. Mat. Pura Appl. (4)32, 19–45(1982) · Zbl 0524.53023
[12] Friedrich Th., Ivanov S. (2002) Parallel spinors and connections with skew-symmetric torsion in string theory. Asian J. Math. 6, 303–336 · Zbl 1127.53304
[13] Friedrich Th., Ivanov S. (2003) Killing spinor equations in dimension 7 and geometry of integrable G 2 manifolds. J. Geom. Phys. 48, 1–11 · Zbl 1029.81037
[14] Gauntlett, J., Kim, N., Martelli, D., Waldram, D.: Fivebranes wrapped on SLAG three-cycles and related geometry. JHEP 0111 018 (2001)
[15] Gauntlett J., Martelli D., Waldram D. (2004) Superstrings with Intrinsic torsion. Phys. Rev. D69:086002
[16] Gibbons G.W., Page D.N., Pope C.N. (1990) Einstein metrics on S 3, \(\mathbb{R}^3\) , and \(\mathbb{R}^4\) bundles. Commun. Math. Phys. 127, 529–553 · Zbl 0699.53053
[17] Gray, A.: Vector cross product on manifolds. Trans. Am. Math. Soc. 141, 463–504 (1969); Correction 148, 625(1970) · Zbl 0182.24603
[18] Ivanov P., Ivanov S. (2005) SU(3)-instantons and G 2, Spin(7)-heterotic string solitons. Commun. Math. Phys. 259, 79–102 · Zbl 1082.53027
[19] Joyce D. (1996) Compact Riemannian 7-manifolds with holonomy G 2. I. J. Diff. Geom. 43, 291–328 · Zbl 0861.53022
[20] Joyce D. (1996) Compact Riemannian 7-manifolds with holonomy G 2. II. J. Diff. Geom. 43, 329–375 · Zbl 0861.53023
[21] Joyce D. (2000) Compact Riemannian manifolds with special holonomy. Oxford, Oxford University Press · Zbl 1027.53052
[22] Kovalev A. (2003) Twisted connected sums and special Riemannian holonomy. J. Reine Angew. Math. 565, 125–160 · Zbl 1043.53041
[23] Lawson B., Michelsohn M.-L. (1989) Spin Geometry. Princeton, NJ, Princeton University Press · Zbl 0688.57001
[24] Leung N.C. (2002) TQFT for Calabi-Yau three folds and G 2 manifolds. Adv. Theor. Math. Phys. 6, 575–591 · Zbl 1043.81065
[25] Salamon, S.: Riemannian geometry and holonomy groups. London: Pitman Res. Notes Math. Ser., 201 1989 · Zbl 0685.53001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.