Ricci curvature of double manifolds via isoparametric foliations

Abstract Given a closed manifold M and a vector bundle ξ of rank n over M , by gluing two copies of the disc bundle of ξ , we can obtain a closed manifold D ( ξ , M ) , the so-called double manifold. In this paper, we firstly prove that each sphere bundle S r ( ξ ) of radius r > 0 is an isoparametric hypersurface in the total space of ξ equipped with a connection metric, and for r > 0 small enough, the induced metric of S r ( ξ ) has positive Ricci curvature under the additional assumptions that M has a metric with positive Ricci curvature and n ≥ 3 . As an application, if M admits a metric with positive Ricci curvature and n ≥ 2 , then we construct a metric with positive Ricci curvature on D ( ξ , M ) . Moreover, under the same metric, D ( ξ , 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. [25] ), we study Ricci curvature of them with isoparametric foliations in the last part.