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May 1, 2014 at 18:58 comment added Yunhyung Cho I see. You mean that the pull-back bundle $i^*TM% over %\mu^{-1}(0)/S^1$ splits into the sum of a tangent bundle over $\mu^{-1}(0)/S^1$ and a trivial line bundle. So, the argument is true for any characteristic classes, not only first Chern class
May 1, 2014 at 7:39 vote accept Yunhyung Cho
May 1, 2014 at 7:40
May 1, 2014 at 5:41 history answered Ben Webster CC BY-SA 3.0