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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 ...

Let $M$ be a (compact) $G$-homogeneous space with fibre group $H$, and let ${\cal E}$ be a $G$-equivariant $k$-dimensional vector bundle over $M$ with corresponding representation $\pi:H \to $R$^k$. ...