Curvature decomposition of G_2 manifolds

Richard Cleyton; Stefan Ivanov

doi:10.1016/j.geomphys.2008.06.002

paper

Curvature decomposition of G_2 manifolds

arXiv:math/0702289 · doi:10.1016/j.geomphys.2008.06.002

Abstract

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 Fernández-Gray type of $G_{2}$ structure whose scalar curvature vanishes may be realized such that the metric has holonomy contained in $G_{2}$ .

LaTeX 2e, 26 pages, 2 tables. Changes in version 2: shortened, reorganized, misprints corrected, several remarks and new introduction. A formula in the proof of Theorem 1.2a has been corrected. Submitted

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