most recent 30 from http://mathoverflow.net 2013-05-20T20:30:25Z http://mathoverflow.net/feeds/user/25240 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/102742/weak-hopf-conjecture Xiaoyang Chen 2012-07-20T14:43:55Z 2012-07-26T20:24:09Z

<p>Hi: I am thinking of the following problem which is related to weak Hopf conjecture:</p> <p>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.</p> <p>Is there anyone who knows any work on this problem? Thanks</p>