Splitting theorems for nonnegatively curved open manifolds with large ideal boundary
Gromov defined the ideal boundary \(M(\infty)\) with the Tits metric \(d_ \infty\) of a nonnegatively curved complete open Riemannian manifold \(M\). It is our purpose to study the relation between the ideal boundary \(M(\infty)\) and the structure of \(M\). We consider the radius of the ideal boundary, which is defined by \(\text{rad } M(\infty):= \inf_{y\in M(\infty)} \sup_{x\in M(\infty)} d_ \infty (x,y)\), and prove the following statement. For any integer \(n\geq 2\) there exists an \(\varepsilon(n)>0\) such that if a nonnegatively curved \(n\)-dimensional complete open Riemannian manifold \(M\) satisfies \(\text{rad } M(\infty)>\pi- \varepsilon(n)\), then \(M\) is isometric to the Riemannian product \(S\times N\), where \(S\) is a compact Riemannian manifold and \(N\) a complete Riemannian manifold diffeomorphic to \(\mathbb{R}^ k\). Note here that \(N\) is not necessarily isometric to \(\mathbb{R}^ k\) and that \(\text{rad } M(\infty)\) is not equal to \(\pi\) in general.
scientific article; zbMATH DE number 549730
scientific article