Warped products with critical Riemannian metric (Q1908715)

Let \((B,g)\) and \((F,\overline{g})\) be two Riemannian manifolds and let \(f\) be a positive smooth function on \(B\). Then the warped product manifold \(M = B \times_f F\) is defined by the Riemannian metric \(\widetilde {g} = \pi^*(g)+ (f\circ \pi)^2 \sigma^* (\overline{g})\), where \(\pi\) and \(\sigma\) are the projections of \(B \times F\) onto \(B\) and \(F\), respectively. Let \(\widetilde{R}\), \(R\) and \(\overline{R}\) be the curvature tensors, and \(\widetilde{\gamma}\), \(\gamma\), and \(\overline{\gamma}\) the scalar curvatures of \(M\), \(B\), and \(F\), respectively. Consider the Riemannian functional \(H(\widetilde{\gamma}) =\int_M \widetilde{\gamma}^2 d\mu\) on \(M\), where \(d \mu\) is the volume element measured by \(\widetilde\gamma\). A critical point of \(H(\widetilde{\gamma})\) is called a critical Riemannian metric on \(M\). In this paper, the author proves that if \(\widetilde{\gamma}\) is a critical Riemannian metric on a warped product manifold \(M\), then \(M\) is the Riemannian product space or \(\widetilde{\gamma}\) is constant on \(M\). Also, it is shown that if \(\widetilde{\gamma} \neq 0\) is a critical Riemannian metric on \(M\), then \(\overline {\gamma}\) on \(F\) is a critical Riemannian metric if and only if \(F\) is Einstein.