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Let $G$ be a Lie group with a closed subgroup $H$, and let $M$ be a smooth $H$-manifold. I am searching for a reference where it is proved that the tangent bundle of $G \times_H M$ is isomorphic to the bundle $TG \times_{TH} TM$ where $TH$ and $TG$ carry the tangent group structures.

I once stumbled upon a paper where this is proved in the beginning, but I cannot find it anymore. Does anybody know which paper I am talking about?

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    $\begingroup$ Aren't there (essentially by the implicit function theorem) a neighbourhood $U_H$ of $1$ in $H$ and a set $U_H'$ containing $1$ in $G$ such that $U_H \times U_H' \to G$ is a diffeomorphism onto a neighbourhood of the identity? $\endgroup$
    – LSpice
    Commented Dec 8, 2022 at 20:17
  • $\begingroup$ This sounds reasonable. It should also not be hard to show that the map $TG \times TM \cong T(G \times M) \to T(G \times_H M)$ factors through $TG \times_{TH} TM$ and then a dimension argument shows that the induced map is in fact an isomorphism. I am however more interested in a (the?) reference. $\endgroup$
    – Lukas
    Commented Dec 9, 2022 at 9:47

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