Vector Bundles with Infinitely Many Souls

We construct the first examples of manifolds, the simplest one being S3×S4×R5, which admit infinitely many complete nonnegatively curved metrics with pairwise nonhomeomorphic souls. According to the soul theorem of J. Cheeger and D. Gromoll [CG72], a complete open manifold of nonnegative sectional curvature is diffeomorphic to the total space of the normal bundle of a compact totally geodesic submanifold, called a soul. The soul is not unique but any two souls are mapped to each other by an ambient diffeomorphism inducing an isometry on the souls [Sha79]. In this note we show that the homeomorphism type of the soul generally depends on the metric; namely the following is true. Theorem 1. There exist infinitely many complete Riemannian metrics on S × S × R with sec ≥ 0 and pairwise nonhomeomorphic souls. The proof applies some classical techniques of geometric topology to recent examples of nonnegatively curved manifolds due to K. Grove and W. Ziller [GZ00]. As we explain below, it is much easier to produce a manifold with finitely many nonnegatively curved metrics having nonhomeomorphic souls, however the full power of [GZ00] is needed to get infinitely many such metrics. Grove and Ziller [GZ00] showed that any principal S×S-bundle over S admits an S × S-invariant metric with sec ≥ 0. By O’Neill’s formula, all associated bundles admit metrics with sec ≥ 0 which gives rise to a rich class of examples, including all sphere bundles over S with structure group SO(4). Note that the souls in Theorem 1 are the total spaces of S-bundles over S with structure group SO(3). Theorem 1 is a particular case of the following. Theorem 2. Let ξ be a rank n vector bundle over S with structure group SO(3), let q : S → S be a smooth Sm−1-bundle with structure group SO(3), and let η be the q-pullback of ξ. If m = 4, n > 4, or if m > 4, n > m+3, then the total space of η admits infinitely many complete Riemannian metrics with sec ≥ 0 and pairwise nonhomeomorphic souls. The main topological tool used in this paper is a result of L. Siebenmann [Sie69] that generalizes the famous Masur’s theorem: any tangential homotopy equivalence of closed smooth n-manifolds is homotopic to a diffeomorphism after taking the Received by the editors June 17, 2001 and, in revised form, September 25, 2001. 2000 Mathematics Subject Classification. Primary 53C20.