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In this paper we review $G_2$ and $Spin(7)$ geometries in relation with aspecial type of metric structure which we call warped-like product metric. Wepresent a general ansatz of warped-like product metric as a definition ofwarped-like product. Considering fiber-base decomposition, the definition ofwarped-like product is regarded as a generalization of multiply-warped productmanifolds, by allowing the fiber metric to be non block diagonal. For somespecial cases, we present explicit example of $(3+3+2)$ warped-like productmanifolds with $Spin(7)$ holonomy of the form $M=F\times B$, where the base $B$is a two dimensional Riemannian manifold, and the fibre $F$ is of the form$F=F_1\times F_2$ where $F_i$'s $(i=1,2)$ are Riemannian $3$-manifolds.Additionally an explicit example of $(3+3+1)$ warped-like product manifold with$G_2$ holonomy is studied. From the literature, some other special warped-likeproduct metrics with $G_2$ holonomy are also presented in the present study. Webelieve that our approach of the warped-like product metrics will be animportant notion for the geometries which use warped and multiply-warpedproduct structures, and especially manifolds with exceptional holonomy.

Warped-like product manifolds with exceptional holonomy groups