Characterizations of twisted product manifolds to be warped product manifolds (Q2850105)

Let \((B,g_{_B})\), \((F,g_{_F})\) be semi-Riemannian manifolds and \(f\), \(b\) positive smooth functions defined on \(B \times F\). The double twisted product \(M=_{f}\!\!B\times _b F\), with twisting functions \(b\) and \(f\), is the manifold \(M=(B \times F, f^2 g_{_B} + b^2 g_{_F})\).NEWLINENEWLINEIf \(f=1\), \(B \times _b F\) is called the twisted product with twisting function \(b\) and if, in addition, the function \(b\) only depends on the points of \(B\), then \(B \times _b F\) is called the warped product of \((B,g_{_B})\) and \((F,g_{_F})\) with warping function \(b\).NEWLINENEWLINEThe authors look for some conditions under which a twisted product can be expressed as a warped product. Such conditions involve the Weyl conformal curvature tensor, via flatness or parallelism, and the Weyl projective tensor. Finally, they extend some results to the case of multiply twisted product manifolds.