Moduli spaces of nonnegative sectional curvature and non-unique souls

Abstract

We apply various topological methods to distinguish connected components of moduli spaces of complete Riemannian metrics of nonnegative sectional curvature on open manifolds. The new geometric ingredient is that souls of nearby nonnegatively curved metrics are ambiently isotopic.

Moduli space of metrics of nonnegative sectional or positive Ricci curvature on homotopy real projective spaces

“…This implies Theorem A. For the convenience of the reader we recall the details from the arguments in [BKS11,DKT18] with the adequate adaptations.…”

Open manifolds with non-homeomorphic positively curved souls

“…This is probably not optimal. All we know is that any pair of souls as in Theorem A necessarily has codimension at least three: according to [BKS11], any two codimension-two souls of a simply connected open manifold are homeomorphic. There is, however, the following result on positively-curved codimension-two souls due to Belegradek, Kwasik and Schultz: Indeed, this is essentially the case m = 0 of Theorem 1.4 in [BKS15]; the exact statement may easily be extracted from the proof of this theorem given there (see page 41).…”

“…The situation is summarized in Figure 1. Here, we present open manifolds with pairs of souls that satisfy all three properties simultaneously: In combination with results of [KPT05,BKS11], Theorem A yields some consequences on the topology of the moduli space of Riemannian metrics with nonnegative sectional curvature on the corresponding spaces. This is explained in Section 6.…”

“…e n s s p a c e s E s c h e n b u r g s p a c e s [B K S 1S 4 [Bel03] S 2 × S 2 × S 3 × S 3 [KPT05]and other examples[BKS11] …”

Tangential thickness of manifolds

“…Finally, Section 8.1 contains some comments and remarks concerning smooth tangential thickness. Some of the techniques and ideas of this paper were applied in [14] and [15] when studying and classifying open complete manifolds of nonnegative curvature (see also [60] for further results on such questions). By the results of J. Cheeger and D. Gromoll [21], such manifolds are diffeomorphic to the total space of a normal bundle to a compact locally geodesic submanifold called a soul.…”