(3+3+2) warped-like product manifolds with Spin(7) holonomy

We consider a generalization of eight-dimensional multiply warped product manifolds as a special warped product, by allowing the fiber metric to be non-block diagonal. We define this special warped product as a (3 + 3 + 2) warped-like manifold of the form M = F x B. where the base B is a two-dimensional Riemannian manifold, and the fibre F is of the form F = F-1 x F-2 where the F-i(i = 1, 2) are Riemannian 3-manifolds. We prove that the connection on M is completely determined by the requirement that the Bonan 4-form given in the work of Yasui and Ootsuka [Y. Yasui and T. Ootsuka, Spin(7) holonomy manifold and superconnection, Class. Quantum Gravity 18(2001)807-816] be closed. Assuming that the F-i are complete, connected and simply connected, it follows that they are isometric to S-3 with constant curvature k > 0 and the Yasui-Ootsuka solution is unique in the class of (3 + 3 + 2) warped-like product metrics admitting a specific Spin(7) structure. Crown Copyright (C) 2011 Published by Elsevier B.V. All rights reserved.