## $$G_2$$-manifolds with parallel characteristic torsion.(English)Zbl 1141.53019

The author studies seven-dimensional Riemannian manifolds $$(M,g)$$ admitting a $$G_2$$ structure $$\varphi$$, which is cocalibrated, i.e., $$d*\varphi=0$$. Then there exists a characteristic connection $$\nabla^c$$ with totally skew symmetric torsion $$T^c=\frac16(d\varphi,*\varphi)-*d\varphi$$ which preserves the $$G_2$$ structure. The author investigates those characteristic connections for which the torsion is $$\nabla^c$$-parallel. In this case the manifold $$M$$ admits a $$\nabla^c$$-parallel spinor field $$\Psi$$ such that $$T^c\cdot\Psi=a\cdot \Psi$$, where the factor $$a$$ depends on an algebraic type of the torsion form.
The author gives a classification of cocalibrated $$G_2$$-manifolds with parallel characteristic torsion and non-abelian holonomy algebra. There are eight non-abelian Lie subalgebras of the Lie algebra $${\mathfrak g}_2$$. The author describes for each of these sub-algebras the set of admissible torsion forms $$T^c$$ and the geometry of the corresponding $$G_2$$-manifolds. For example, a $$G_2$$-manifold with parallel torsion and $${\mathfrak{hol}}(\nabla^c)= {\mathfrak{su}}(3)$$ turns out to be homothetic to an $$\eta$$-Einstein Sasakian manifold or a product of a nearly Kähler 6-manifold by $$\mathbb{R}$$. A regular $$G_2$$-manifold with parallel torsion and $${\mathfrak{hol}}(\nabla^c)= {\mathfrak u}(2)$$ is proved to be a principal $$S^1$$-bundle over the projective space $$\mathbb{CP}^3$$ or a flag manifold $$\mathbb{F}(1,2)$$ equipped with their standard nearly Kähler structures coming from the twistor construction or a principal $$S^1$$-bundle over the Kähler manifold satisfying some further conditions. In the case $${\mathfrak{hol}}(\nabla^c)= {\mathfrak{so}}_{ir}(3)$$ the author gives the complete classification result: A complete, simply-connected and cocalibrated $$G_2$$-manifold with parallel characteristic torsion and $${\mathfrak{hol}}(\nabla^c)= {\mathfrak{so}}_{ir}(3)$$ is isometric to $$\text{SO}(5)/\text{SO}_{ir}(3)$$. Also in the case $${\mathfrak{hol}}(\nabla^c)= {\mathfrak{su}}_c(2)$$ the author gives a complete classification.
It should be noted that $$G_2$$-manifolds with a parallel characteristic torsion are solutions of the common sector of type II superstring theory.

### MSC:

 53C10 $$G$$-structures 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
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### References:

 [1] Agricola, I.; Friedrich, Th., On the holonomy of connections with skew-symmetric torsion, Math. ann., 328, 711-748, (2004) · Zbl 1055.53031 [2] Agricola, I.; Friedrich, Th., The Casimir operator of a metric connection with skew-symmetric torsion, J. geom. phys., 50, 188-204, (2004) · Zbl 1080.53043 [3] Agricola, I.; Friedrich, Th.; Nagy, P.-A.; Puhle, C., On the Ricci tensor in the common sector of type II string theory, Class. quantum grav., 22, 2569-2577, (2005) · Zbl 1074.81056 [4] Alexandrov, B.; Friedrich, Th.; Schoemann, N., Almost Hermitian 6-manifolds revisited, J. geom. phys., 53, 1-30, (2005) · Zbl 1075.53036 [5] Belgun, F.; Moroianu, A., Nearly Kähler 6-manifolds with reduced holonomy, Ann. global anal. geom., 19, 307-319, (2001) · Zbl 0992.53037 [6] Besse, A.L., Einstein manifolds, (1987), Springer-Verlag Berlin, Heidelberg · Zbl 0613.53001 [7] Boyer, C.; Galicki, K.; Matzeu, P., On eta-Einstein Sasakian geometry, Comm. math. phys., 262, 177-208, (2006) · Zbl 1103.53022 [8] Chiossi, S.; Salamon, S., The intrinsic torsion of $$\operatorname{SU}(3)$$ and $$\operatorname{G}_2$$-structures, (), 115-133 · Zbl 1024.53018 [9] Cleyton, R.; Swann, A., Einstein metrics via intrinsic or parallel torsion, Math. Z., 247, 513-528, (2004) · Zbl 1069.53041 [10] Dynkin, E.B., The maximal subgroups of classical groups, Tr. mosk. mat. obshch., 1, 39-166, (1952) [11] Fernandez, M.; Gray, A., Riemannian manifolds with structure group $$\operatorname{G}_2$$, Annali di math. pura e appl., 132, 19-45, (1982) · Zbl 0524.53023 [12] Friedrich, Th., On types of non-integrable geometries, Rend. circ. mat. di Palermo, 71, 99-113, (2003) · Zbl 1079.53041 [13] Friedrich, Th.; Ivanov, S., Parallel spinors and connections with skew-symmetric torsion in string theory, Asian J. math., 6, 303-336, (2002) · Zbl 1127.53304 [14] Friedrich, Th.; Kath, I.; Moroianu, A.; Semmelmann, U., On nearly parallel $$\operatorname{G}_2$$-structures, J. geom. phys., 23, 256-286, (1997) · Zbl 0898.53038 [15] Friedrich, Th.; Kim, E.C., The einstein – dirac equation on Riemannian spin manifolds, J. geom. phys., 33, 128-172, (2000) · Zbl 0961.53023 [16] Kostant, B., A cubic Dirac operator and the emergence of Euler number multiplets of representations for equal rank subgroups, Duke math. J., 100, 447-501, (1999) · Zbl 0952.17005 [17] N. Schoemann, Almost Hermitian structures with parallel torsion, J. Geom. Phys., in press · Zbl 1137.53014
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