Hi: I am thinking of the following problem which is related to weak Hopf conjecture:
Let $E$ be the total space of a vector bundle over a compact nonnegatively curved manifold $B$. Let k=the rank of $E$ and k > dim B. Now assume $g$ is any complete metric on E with nonnegative sectional curvature. By soul theorem. there is a soul $N$ inside $E$. Also by a theorem of Guijarro and Walschap, for samll $r>0$, the normal sphere bundle $N_r$ is also nonnegatively curved with respect to the induced metric of $g$. Now the question is: Show this induced metric on $N_r$ can not have positive sectional curvature for all sufficiently small r > 0! If E is the trivial R^3 bundle over S^2, it is the so called "weak Hopf conjecture" Proposed by Gromoll and Tapp.
Is there anyone who knows any work on this problem? Thanks