Can someone help me to visualize geometrically the fiber bundle $U(n-1)\rightarrow U(n) \rightarrow S^{2n-1}$, what are the open sets where it trivializes?
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4A fiber bundle trivializes over any contractible open set, so any bundle over $S^k$ trivializes over the complement of a point (and therefore over any proper subset). – Jun 22 '16 at 05:02
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For purposes of intuition, it's probably best 1. To understand homogeneous spaces in general; 2. To think of $S^{2n-1}$ as the unit sphere in $\mathbf{C}^{n}$. It's quite clear that $U(n)$ acts transitively, and the stabilizer of a point is isomorphic to $U(n-1)$. – Andrew D. Hwang Jun 25 '16 at 17:31