Fiber structures of special (4+3+1) warped-like manifolds with Spin(7) holonomy


INTERNATIONAL JOURNAL OF GEOMETRIC METHODS IN MODERN PHYSICS, vol.11, no.8, 2014 (SCI-Expanded) identifier identifier


L generalization of 8-dimensional multiply-warped product manifolds is considered as a special warped product, by allowing the fiber metric to be non-block diagonal. Motivating from the previous paper [S. Uguz and A. H. Bilge, (3 + 3 + 2) warped-like product manifolds with Spin(7) holonomy, J. Geom. Phys. 61 (2011) 1093-1103], we present a special warped product as a (4 + 3 + 1) warped-like manifold of the form M = F x B, where the base B is a 1-dimensional Riemannian manifold, and the fiber F is of the form F = F-1 x F-2 where F-i's (i = 1, 2) are Riemannian 4- and 3-manifolds, respectively. It is showed that the connection on M is entirely determined provided that the Bonan 4-form is closed. Assuming that the F-i's are complete, connected and simply connected, it is proved that the 3-dimensional fiber is isometric to S-3 with constant curvature k > 0. Finally, the geometric properties of the 4-dimensional fiber of M are studied.