Curvature decomposition of G_2 manifolds

Details Curvature decomposition of G_2 manifolds Curvature decomposition of G_2 manifolds

2007-02-10

arxiv.org/abs/math/0702289v2

Mathematics

Differential Geometry

LaTeX 2e, 26 pages, 2 tables. Changes in version 2: shortened, reorganized, misprints corrected, several remarks and new intro

Scientific paper

10.1016/j.geomphys.2008.06.002

Explicit formulas for the $G_2$-components of the Riemannian curvature tensor on a manifold with a $G_2$ structure are given in terms of Ricci contractions. We define a conformally invariant Ricci-type tensor that determines the 27-dimensional part of the Weyl tensor and show that its vanishing on compact $G_2$ manifold with closed fundamental form forces the three-form to be parallel. A topological obstruction for the existence of a $G_2$ structure with closed fundamental form is obtained in terms of the integral norms of the curvature components. We produce integral inequalities for closed $G_2$ manifold and investigate limiting cases. We make a study of warped products and cohomogeneity-one $G_2$ manifolds. As a consequence every Fern\'andez-Gray type of $G_2$ structure whose scalar curvature vanishes may be realized such that the metric has holonomy contained in $G_2$.

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