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I am thinking of the following problem which is related to the weak Hopf conjecture:

Let $E$ be the total space of a vector bundle over a compact nonnegatively curved manifold $B$. Let $k$ be the rank of $E$ and suppose that $k > \dim B$. Now assume that $g$ is a complete metric on $E$ with non-negative sectional curvature. By the soul theorem. there is a soul $N$ inside $E$. Also by a theorem of Guijarro and Walschap, for small $r>0$, 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 $\Bbb R^3$ bundle over $\Bbb S^2$, it is the so called "weak Hopf conjecture" Proposed by Gromoll and Tapp.

Is there anyone who knows any work on this problem?

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    $\begingroup$ But if I have some ideas about this, why would I want to share it on MathOverflow? Maybe you should ask your question a little differently and ask for recent work or progress on this question. $\endgroup$
    – Deane Yang
    Jul 20, 2012 at 15:11
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    $\begingroup$ see Kris Tapp's recent paper front.math.ucdavis.edu/1109.5150 $\endgroup$ Jul 26, 2012 at 20:03

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