Geodesic
Cocalibrated $G_2$-manifolds with Ricci flat characteristic connection
Communications in Mathematics, Tome 21 (2013) no. 1, pp. 1-13.

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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.
Classification : 53C25, 81T30
Keywords: cocalibrated $G_2$-manifolds; connections with torsion 
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     author = {Friedrich, Thomas},
     title = {Cocalibrated $G_2$-manifolds with {Ricci} flat characteristic connection},
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Friedrich, Thomas. Cocalibrated $G_2$-manifolds with Ricci flat characteristic connection. Communications in Mathematics, Tome 21 (2013) no. 1, pp. 1-13. http://geodesic.mathdoc.fr/item/COMIM_2013__21_1_a0/