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

• Autores: Richard Cleyton, Stefan Ivanov
• Localización: Mathematical research letters, ISSN 1073-2780, Vol. 15, Nº 4, 2008, págs. 631-640
• Idioma: inglés
• DOI: 10.4310/mrl.2008.v15.n4.a3
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• Resumen

  • For $G_2$-manifolds the Fern\'andez-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 K\"ahler manifolds of class $\mathcal W_4$ and conformal transformations of nearly K\"ahler 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 K\"ahler manifold, respectively. The properties of the Riemannian curvature of the spaces under consideration are also explored.