Conformal equivalence between certain geometries in dimension 6 and 7

Details Conformal equivalence between certain geometries in dimension 6 and 7 Conformal equivalence between certain geometries in dimension 6 and 7

2006-07-20

arxiv.org/abs/math/0607487v2

Mathematics

Differential Geometry

8 pages, typos and a sign inconsistency has been fixed

Scientific paper

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

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