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.