2016/01/13 by Chiakuei Peng, Chao Qian, Peng, ChiaKuei +1
Mathematics · Medicine · #Advanced Neuroimaging Techniques and Applications #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.1601.03125
openalex publication_date 2016/01/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Sr(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 Sr(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 Sn+1+ and Sn+1-, one can get double manifolds D(Sn+1+) and D(Sn+1-). Inspired by Tang, Xie and Yan's work on scalar curvature of such manifolds with isoparametric foliations(cf. \citeTXY12), we study Ricci curvature of them with isoparametric foliations in the last part.