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We find a remarkable family of G2 structures defined on certain principal SO(3)- bundles P± → M associated with any given oriented Riemannian 4-manifold M. Such structures are always cocalibrated. The study starts with a recast of the Singer-Thorpe equations of 4-dimensional geometry. These are applied to the Bryant-Salamon construction of complete G2- holonomy metrics on the vector bundle of self- or anti-self-dual 2-forms on M. We then discover new examples of that special holonomy on disk bundles over H4 and H²C, respectively, the real and complex hyperbolic space. Only in the end we present the new G2 structures on principal bundles. [ABSTRACT FROM AUTHOR]

Given a Riemannian metric on a compact smooth manifold, we consider its Schouten tensor, which is a tensor field of type (0, 2) arising in the remainder of the Weyl part in the standard decomposition of the curvature tensor of the metric. We study extremal properties of the Schouten functional, defined to be the scaling-invariant L-norm of the Schouten tensor. It is proved, for instance, that space form metrics are characterized as critical points of the Schouten functional among conformally flat metrics. [ABSTRACT FROM AUTHOR]

Abstract: A local description of proper Hermitian–Walker structures that are locally conformally Kähler, self-dual, ∗-Einstein or Einstein is given. This is used to supply examples of indefinite Einstein strictly almost Hermitian structures showing that an integrability result in [K.-D. Kirchberg, Integrability conditions for almost Hermitian and almost Kähler 4-manifolds, arXiv:math.DG/0605611] does not hold for metrics of signature . [Copyright &y& Elsevier]

Abstract: It is shown that any proper almost Hermitian structure on a Walker 4-manifold is isotropic Kähler. Moreover, a local description of proper almost Kähler structures that are self-dual, -Einstein or Einstein is given and it is proved that any proper strictly almost Kähler Einstein structure is self-dual, Ricci flat and -Ricci flat. This is used to supply examples of flat indefinite non-Kähler almost Kähler structures. [Copyright &y& Elsevier]

Abstract: We construct infinitely many seven-dimensional Einstein metrics of weak holonomy . These metrics are defined on principal SO(3) bundles over four-dimensional Bianchi IX orbifolds with the Tod–Hitchin metrics. The Tod–Hitchin metric has an orbifold singularity parametrized by an integer, and is shown to be similar near the singularity to the Taub-NUT de Sitter metric with a special charge. We show, however, that the seven-dimensional metrics on the total space are actually smooth. The geodesics on the weak manifolds are discussed. It is shown that the geodesic equation is equivalent to the Hamiltonian equation of an interacting rigid body system. We also discuss M-theory on the product space of AdS4 and the seven-dimensional manifolds, and the dual gauge theories in three dimensions. [Copyright &y& Elsevier]