I want to know which isomorphism of vector bundles is the following?


[alt text http://2.bp.blogspot.com/\_uGcgLiQvkI8/TAgpIivl6oI/AAAAAAAAAKk/DCg9iK2W3rA/s1600/Capture-52.png](http://2.bp.blogspot.com/_uGcgLiQvkI8/TAgpIivl6oI/AAAAAAAAAKk/DCg9iK2W3rA/s1600/Capture-52.png)


where $G/H$ is the quotient of group $G$ by its subgroup $H$ and $T(G/H)$ is the tangent bundle, $G\times\_H \mathfrak{g}/\mathfrak{h}$ is the bundle associated to the principal bundle $G\to G/H$ via the adjointe representation on $\mathfrak{g}/\mathfrak{h}$



A metric on T(G/H)
There is an obvious map $G\times\mathfrak g\to G\times\_H\mathfrak g/\mathfrak h$, and an isomorphism $TG\to G\times\mathfrak g$. On the other hand, the projection $G\to G/H$ gives a map $TG\to T(G/H)$. Now you can construct a map $T(G/H)\to G\times\_H\mathfrak g/\mathfrak h$ as the composition $$T(G/H)\leftarrow TG\to G\times\mathfrak g\to G\times\_H\mathfrak g/\mathfrak h.$$ Here the backwards arrow means "pick any preimage".


