Special Warped-Like Product Manifolds with (Weak) $G_2$ Holonomy

Abstract

By using the fiber-base decompositions of manifolds, the definition of warped-like product is regarded as a generalization of multiply warped product manifolds, by allowing the fiber metric to be not block diagonal. We consider the (3 + 3 + 1) decomposition of 7-dimensional warped-like product manifolds, which is called a special warped-like product of the form $M = F  × B$; where the base $B$ is a onedimensional Riemannian manifold and the fiber $F$ has the form $F = F_1 × F_2$ where $F_i ; i = 1, 2$, are Riemannian 3-manifolds. If all fibers are complete, connected, and simply connected, then they are isometric to $S_3$ with constant curvature $k > 0$ in the class of special warped-like product metrics admitting the (weak) $G_2$ holonomy determined by the fundamental 3-form.