Remark on Laplacians and Riemannian Submersions with Totally Geodesic Fibers

Kazumasa Narita

Published 2024-11-26Version 1

Given a Riemannian submersion $(M,g) \to (B,j)$ each of whose fiber 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. We prove that if each fiber is Einstein and $(M,g)$ satisfies a certain condition about its Ricci curvature, then the $l$-th positive eigenvalue $\lambda_{l}(g_{t})$ of the Laplacian on $(M,g_{t})$ is bounded from below by a positive constant independent of $l$. This in particular implies that the scale-invariant quantity $\lambda_{1}(g_{t})\mbox{Vol}(M,g_{t})^{2/\mbox{dim}M}$ goes to $\infty$ with $t$. We consider many examples. In particular, we consider Riemannian submersions from compact rank one symmetric spaces and the twistor fibration of a quaternionic K\"{a}hler manifold of positive scalar curvature.