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Let $\mathbb{R}P^2$ be the lines in $\mathbb{R}^3$ passing through the origin. Let $SO(3)$ act on $\mathbb{R}^3$ canonically. Then $SO(3)$ has an induced action on $\mathbb{R}P^2$. Let $F(\mathbb{R}P^...

Let $G$ be a compact Lie group which act on a manifold $M$. We fix this action throughout our question. Assume that $E\to M$ is a vector bundle which has the potential of admitting ...