Ricci curvature of double manifolds via isoparametric foliations

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 Yⁿ in Sⁿ⁺¹(1), which separates Sⁿ⁺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.