Warped products with critical Riemannian metrics. II (Q2782762)

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, \(\widetilde{S}\), \(S\) and \(\overline{S}\) the Ricci curvature tensors, and \(\widetilde{K}\), \(K\), and \(\overline{K}\) the scalar curvatures of \(M\), \(B\), and \(F\), respectively. Consider the Riemannian functional \(B_M(G)=\int_M \widetilde{K}^2 d\mu\) on \(M\), where \(d \mu\) is the volume element measured by \(\widetilde K\). A critical point of \(B_M(\widetilde{K})\) is called a critical Riemannian metric on \(M\). In part I of this paper [Proc. Japan Acad., Ser. A 71, No. 6, 117-118 (1995; Zbl 0854.53034)] the second author showed that if \(G_B\) is a critical Riemannian metric of the Riemannian functional \(B_M(G)\) for the scalar curvature \(\widetilde{K}\) on \(M\), then the warped product manifold \(M\) is the Riemannian product space or \(\overline{K}\) on \(F\) is constant on \(M\). NEWLINENEWLINENEWLINEIn this paper, the authors consider two Riemannian functionals \(C_M(G)= \int_M \widetilde{S} d\mu\) and \(D_M(G) =\int_M \widetilde{R}^2 d\mu\) on the warped product manifold \(M\), and they prove that for compact orientable manifolds \(B\) and \(F\), if \(G_C\) and \(G_D\) are critical Riemannian metrics of the functionals \(C_M(G)\) and \(D_M(G)\), respectively, then the warped product manifold \(M=B \times_f F\), in both cases, is the Riemannian product space or \(\overline{K}\) on \(F\) is constant on \(M\).