Conformal equivalence between certain geometries in dimension $\mathbf 6$ and $\mathbf 7$

For $G_2$-manifolds the Fernández-Gray class $\mathcal X_1+\mathcal X_4$ is shown to consist of the union of the class $\mathcal X_4$ of $G_2$-manifolds locally conformal to parallel $G_2$-structures and that of conformal transformations of nearly parallel or weak holonomy $G_2$-manifolds of type $\mathcal X_1$. The analogous conclusion is obtained for Gray-Hervella class $\mathcal W_1+\mathcal W_4$ of real $6$-dimensional almost Hermitian manifolds: this sort of geometry consists of locally conformally Kähler manifolds of class $\mathcal W_4$ and conformal transformations of nearly Kähler manifolds in class $\mathcal W_1$. A corollary of this is that a compact $\SU(3)$-space in class $\mathcal W_1+\mathcal W_4$ or $G_2$-space of the kind $\mathcal X_1+\mathcal X_4$ has constant scalar curvature if only if it is either a standard sphere or a nearly parallel $G_2$ or nearly Kähler manifold, respectively. The properties of the Riemannian curvature of the spaces under consideration are also explored.

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Published 1 January 2008