Isomorphism of space of horizontal vectors of $F\left(M\right)$

Question

Let $\left(F\left(M\right),M,\pi\right)$ be the frame bundle of $M$. I am taking an element of $F\left(M\right)$ to be a pair $\left(p,u\right)$ where $p\in M$ and $u:\mathbb{R}^n\to T_pM$ is a linear isomorphism. Let $H_{\left(p,u\right)}F\left(M\right)$ be the space of horizontal tangent vectors of $T_{\left(p,u\right)}F\left(M\right)$. $\pi$ induces an isomorphism $\pi^*:H_{\left(p,u\right)}F\left(M\right)\to T_pM$, but what is $\pi^*$?

Answer 1

The differential $D_{(p,u)} \pi$ is a map $T_{(p,u)} FM \to T_p M$. The map $\pi^*$ is just the restriction of this to the horizontal subspace $H_{(p,u)} FM$, and the definition of a horizontal bundle guarantees it will be an isomorphism (since the kernel of $D \pi$ is the vertical bundle).

Answer 2

The space $H_{(p,u)}FM$ is not well-defined. There is not a canonical horizontal bundle.To each bundle $H$ supplementary to the vertical bundle is associated a connection in the sense of Ehreshman, and the restriction of the differential of $\pi$ to $H$ is an isomorphism.