Given a Riemannian submersion \((M,g) \rightarrow (B,j)\) each of whose fibers is connected and totally geodesic, we consider a certain 1-parameter family of Riemannian metrics \((g_{t})_{t > 0}\) on M, which is called the canonical variation. Let \(\lambda _{1}(g_{t})\) be the first positive eigenvalue of the Laplace–Beltrami operator \(\Delta ^{M}_{g_{t}}\) and \(\hbox {Vol}(M,g_{t})\) the volume of \((M, g_{t})\) . In Bérard-Bergery and Bourguignon (Ill J Math 26(2):181–200,1982 ) showed that the scale-invariant quantity \(\lambda _{1}(g_{t})\hbox {Vol}(M,g_{t})^{2/\hbox {dim}M}\) goes to 0 with t. In this paper, we show that if each fiber is Einstein and (M, g) satisfies a certain condition about its Ricci curvature, then bounds for \(\lambda _{1}(g_{t})\) can be obtained. In particular this implies that \(\lambda _{1}(g_{t})\hbox {Vol}(M,g_{t})^{2/\hbox {dim}M}\) goes to \(\infty \) with t. Moreover, using these bounds, we consider stability of \(g_{t}\) as a critical point of the Yamabe functional.
