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# Article

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Keywords:
cocalibrated \$G_2\$-manifolds; connections with torsion
Summary:
Any 7-dimensional cocalibrated \$G_2\$-manifold admits a unique connection \$\nabla\$ with skew symmetric torsion (see [8]). We study these manifolds under the additional condition that the \$\nabla\$-Ricci tensor vanish. In particular we describe their geometry in case of a maximal number of \$\nabla\$-parallel vector fields.
References:
[1] Agricola, I., Ferreira, A.C.: Einstein manifolds with skew torsion. to appear.
[2] Agricola, I., Friedrich, Th.: On the holonomy of connections with skew-symmetric torsion. Math. Ann., 328, 2004, 711-748, DOI 10.1007/s00208-003-0507-9 | MR 2047649 | Zbl 1055.53031
[3] Agricola, I., Friedrich, Th.: The Casimir operator of a metric connection with skew-symmetric torsion. J. Geom. Phys., 50, 2004, 188-204, DOI 10.1016/j.geomphys.2003.11.001 | MR 2078225 | Zbl 1080.53043
[4] Agricola, I., Friedrich, Th.: A note on flat connections with antisymmetric torsion. Diff. Geom. its Appl., 28, 2010, 480-487, DOI 10.1016/j.difgeo.2010.01.004 | MR 2651537
[5] Apostolov, V., Armstrong, J., Draghici, T.: Local rigidity of certain classes of almost Kähler 4-manifolds. Math. Ann., 323, 2002, 633-666, DOI 10.1007/s002080200319 | MR 1921552 | Zbl 1032.53016
[6] Apostolov, V., Draghici, T., Moroianu, A.: A splitting theorem for Kähler manifolds whose Ricci tensors have constant eigenvalues. Internat. J. Math., 12, 2001, 769-789, DOI 10.1142/S0129167X01001052 | MR 1850671 | Zbl 1111.53303
[7] Friedrich, Th.: G\$_2\$-manifolds with parallel characteristic torsion. J. Diff. Geom. Appl., 25, 2007, 632-648, DOI 10.1016/j.difgeo.2007.06.010 | MR 2373939 | Zbl 1141.53019
[8] Friedrich, Th., Ivanov, S.: Parallel spinors and connections with skew-symmetric torsion in string theory. Asian J. Math., 6, 2002, 303-336, MR 1928632 | Zbl 1127.53304
[9] Friedrich, Th., Ivanov, S.: Killing spinor equation in dimension 7 and geometry of integrable G\$_2\$-manifolds. J. Geom. Phys., 48, 2003, 1-11, DOI 10.1016/S0393-0440(03)00005-6 | MR 2006222
[10] Grantcharov, D., Grantcharov, G., Poon, Y.S.: Calabi-Yau connections with torsion on toric bundles. J. Differential Geom., 78, 2008, 13-32, MR 2406264 | Zbl 1171.53044
[11] LeBrun, C.: Explicit self-dual metrics on CP2 # ... # CP2. J. Differential Geom., 34, 1991, 223-253, MR 1114461

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