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Let $\pi:P\to M$ be a principal bundle with group $G=\pi^{-1}(p)$, and let $u\in P$ and $p=\pi(u)$.

As I understand it, the choice of the vertical tangent space $V_uP=\mathrm{ker}(\pi_*)$ is natural, while there's no natural choice of a horizontal subspace $H_uP$ such that $T_uP=H_uP\oplus V_uP$. A choice of such a $H_uP$ amounts to choosing an Ehresmann connection $\omega\in\mathfrak{g}\otimes T^*P$.

Let's pick a local trivialization $\phi_i:M\times G\to P$ and then define a map $\psi_i:M\to P$ with $\psi_i(x):=\phi_i(x, e)$, where $e\in G$ is the identity element. In other words, we're choosing the local section of points that correspond to the identity element in the local trivialization. The pushforward ${\psi_i}_*$ maps $T_pM$ to a subspace of $T_uP$, and we can define $H_uP$ as the image of ${\psi_i}_*$.

Is that possible? Do we get a (local) Ehresmann connection this way?

Why isn't it a natural choice?

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The point is that the local trivialization is defined on an open subset of $M$ not on $M$ unless the principal bundle is trivial. Yes, you can use a trivialization $(U_i)_{i\in I}$ where $\pi^{-1}(U_i)\simeq U_i\times G$ of the principal bundle, define the connection like that on $U_i\times G$ and use a partition of the unity to glue and obtain a connection on $P$.  

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