Conformal equivalence between certain geometries in dimension 6 and 7

Abstract

For G_2-manifolds the Fernández-Gray class X_1+X_4 is shown to consist of the union of the class 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 X_1. The analogous conclusion is obtained for Gray-Hervella class W_1+W_4 of real 6-dimensional almost Hermitian manifolds: this sort of geometry consists of locally conformally Kähler manifolds of class W_4 and conformal transformations of nearly Kähler manifolds in class W_1. A corollary of this is that a compact SU(3)-space in class W_1+W_4 or G_2-space of the kind X_1+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.

8 pages, typos and a sign inconsistency has been fixed

References in corpus (4)

• Special almost Hermitian geometry
• Curvature decomposition of G_2 manifolds
• Geometric structures of vectorial type
• Espace de twisteurs d'une variete presque hermitienne de dimension 6

Cited by in corpus (8)

• Short-time behaviour of a modified Laplacian coflow of G2-structures
• Prolongations of Lie algebras and applications
• Curvature decomposition of G_2 manifolds
• Einstein warped G2 and Spin(7) manifolds
• Conformally Einstein Products and Nearly Kähler Manifolds
• A Nearly Quaternionic Structure on SU(3)
• G2-structure deformations and warped products
• Geometric Realizability of Covariant Derivative Kähler Tensors for almost Pseudo-Hermitian and almost Para-Hermitian Manifolds