Ricci curvature of double manifolds via isoparametric foliations

TL;DR

This paper investigates Ricci curvature properties of double manifolds constructed from vector bundles over manifolds with positive Ricci curvature, demonstrating the existence of metrics with positive Ricci curvature and natural isoparametric foliations.

Contribution

It proves that sphere bundles are isoparametric hypersurfaces with positive Ricci curvature under certain conditions and constructs positive Ricci curvature metrics on double manifolds with isoparametric foliations.

Findings

Sphere bundles are isoparametric hypersurfaces with positive Ricci curvature for small radii.

Double manifolds admit metrics with positive Ricci curvature if the base manifold does.

Double manifolds can be equipped with natural 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.