Riemannian submersions of open manifolds which are flat at infinity

Abstract

We prove that a base $B^{n-k}$ of a Riemannian submersion $\pi : {M^n} \to B^{n-k}$ is flat, if $M^n$ is flat at infinity and $B^{n-k}$ is compact. As a corollary we obtain a topological gap-phenomenon for open manifolds of nonnegative sectional curvature (Eschenburg-Schroeder-Strake conjecture).