Ricci curvature of double manifolds via isoparametric foliations

Abstract

Given a closed manifold $M$ and a vector bundle $\xi$ of rank $n$ over $M$, by gluing two copies of the disc bundle of $\xi$, we can obtain a closed manifold $D(\xi, M)$, the so-called double manifold. In this paper, we firstly prove that each sphere bundle $S_r(\xi)$ of radius $r>0$ is an isoparametric hypersurface in the total space of $\xi$ equipped with a connection metric, and for $r>0$ small enough, the induced metric of $S_r(\xi)$ has positive Ricci curvature under the additional assumptions that $M$ has a metric with positive Ricci curvature and $n\geq3$. As an application, if $M$ admits a metric with positive Ricci curvature and $n\geq2$, then we construct a metric with positive Ricci curvature on $D(\xi, M)$. Moreover, under the same metric, $D(\xi, M)$ admits a natural isoparametric foliation. For a compact minimal isoparametric hypersurface $Y^{n}$ in $S^{n+1}(1)$, which separates $S^{n+1}(1)$ into $S^{n+1}_+$ and $S^{n+1}_-$, one can get double manifolds $D(S^{n+1}_+)$ and $D(S^{n+1}_-)$. Inspired by Tang, Xie and Yan's work on scalar curvature of such manifolds with isoparametric foliations(cf. \cite{TXY12}), we study Ricci curvature of them with isoparametric foliations in the last part.