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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

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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. –  Deane Yang Jul 20 '12 at 15:11
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see Kris Tapp's recent paper front.math.ucdavis.edu/1109.5150 –  Igor Belegradek Jul 26 '12 at 20:03
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