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Special warped-like product manifolds with (weak) \(G_{2}\) holonomy. (English) Zbl 1328.53061

Ukr. Math. J. 65, No. 8, 1257-1272 (2014) and Ukr. Mat. Zh. 65, No. 8, 1126-1140 (2013).
Summary: 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 \(M=F\times B\); where the base \(B\) is a one dimensional Riemannian manifold and the fiber \(F\) has the form \(F=F_1\times F_2\) 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.

MSC:

53C29 Issues of holonomy in differential geometry
53C20 Global Riemannian geometry, including pinching
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