Ricci curvature of double manifolds via isoparametric foliations

Abstract

Given a closed manifold $M$ and a vector bundle $\hat{I}¾$ of rank $n$ over $M$, by gluing two copies of the disc bundle of $\hat{I}¾$, we can obtain a closed manifold $D(\hat{I}¾,M)$, the so-called double manifold. In this paper, we firstly prove that each sphere bundle $S_r(\hat{I}¾)$ of radius $r>0$ is an isoparametric hypersurface in the total space of $\hat{I}¾$ equipped with a connection metric, and for $r>0$ small enough, the induced metric of $S_r(\hat{I}¾)$ has positive Ricci curvature under the additional assumptions that $M$ has a metric with positive Ricci curvature and $n\ge 3$. As an application, if $M$ admits a metric with positive Ricci curvature and $n\ge 2$, then we construct a metric with positive Ricci curvature on $D(\hat{I}¾,M)$. Moreover, under the same metric, $D(\hat{I}¾,M)$ 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.

11 pages