# Relation between connection and connection form

In the smooth setting, we define a connection in a principal $$G$$-bundle $$(P, \pi, M)$$ as a smooth assignment to each point $$p \in P$$ of a subspace $$H_p P$$ of $$T_p P$$ such that the following two conditions hold:

1. $$T_p P \cong V_p P \oplus H_pP$$,
2. $$\delta_{g^*}(H_pP) = H_{p \cdot g}P$$,

where $$V_pP = \{ \tau \in T_pP | \pi_* \tau = 0\}$$ is called the vertical subspace of $$T_pP$$ and $$\delta_g(p) = p \cdot g$$ denotes the right action of $$G$$ on $$P$$.

There is an equivalent formulation based on differential forms. Given an element $$A \in \mathrm{Lie}(G) \cong T_e G$$, we can define a vector field $$X^A$$ defined by $$X_p^A(f) = \frac{d}{dt}_{\Big|t=0} f(p \cdot \mathrm{exp}(tA)).$$ A connection form $$\omega$$ is a $$\mathrm{Lie}(G)$$-valued one-form on $$P$$ such that the following hold:

1. $$\omega_p(X^A) = A, \quad \forall p\in P, \forall A \in \mathrm{Lie}(G)$$,
2. $$\delta_g^*\omega = \mathrm{Ad}_{g^{-1}}\omega, \quad \forall g \in G$$.

I want to prove the following theorem : There is a one-to-one correspondence between connections and connection forms. Given a connection form $$\omega$$, the hint given is to set $$H_pP = \mathrm{ker}(\omega)$$. I do not see why this gives a connection.

Hint: The main point that seems to be missing in your descirption is that the map $$A\mapsto X^A_p$$ defines a linear isomorphism from the Lie algebra of $$G$$ onto the subspace $$V_pP$$ (which usually is called the vertical subspace and not the horizontal subspace). This is just an infinitesimal version of the fact that the orbites of the principal right action $$\delta$$ are the fibers of $$G$$. Using this, you conclude that $$\omega_p$$ restricts to a linear isomoprhism on $$V_pP$$ and then the hint you already got should help.
• Yeah I meant Vertical subspace : edited. Now with this isomorphism in mind, this does the job since $\pi_*(X_p^A) = \frac{d}{dt}_{|t=0} \pi(p \cdot \mathrm{exp}(tA)) = \frac{d}{dt}_{|t=0} \pi(p) = 0$, using the fact that $\pi$ is fiber preserving. The second condition follows from the second condition of the connection form. Thanks a lot! Commented Apr 1, 2021 at 9:28