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A Lie group $G$ acts smoothly, properly and freely on a smooth manifold $M$. We define $S=\cup_{p\in M}T_p(G\cdot p)\subset TM$. Is $S$ a sub vector bundle of $TM\to M$?

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Yes. The projection onto the quotient $p\colon M\to M/G$ induces a morphism of vector bundles $TM\to p^*T(M/G)$ and the bundle you describe is the kernel of this map.

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