Geodesic
Cocalibrated $G_2$-manifolds with Ricci flat characteristic connection
Communications in Mathematics, Tome 21 (2013) no. 1, pp. 1-13 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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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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/

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