Connection 1-form on a principal bundle, uniqueness of the separation of tangent space? I have trouble understanding how a connection one-form can separate and tangent space $T_u$ of a principal bundle uniquely into horizontal and vertical spaces ${H_u}P \oplus {V_u}P$ since from the literature I am learning (mainly Nakahara' book), the connection one-form is a Lie-algebra-valued one-form that satisfies
1,  $\omega ({A^\# }) = A$
2, ${R_{g * }}{H_u}P = {H_{ug}}P$
where ${A^\# }$ is the fundamental vector field.
My question is how does it separate  ${T_u}P$ uniquely since from the first requirement we only project the vertical space into its corresponding Lie algebra. Should we have some additional requirement like $\omega (X) = 0$ for $X \in {H_u}P$??? Or this condition can just be derived from the second requirement?
 A: Your definition is a sort of hybrid of two standard definitions of connection 1-form.  These are:


*

*A connection 1-form is a Lie-algebra valued 1-form $\omega$ which satisfies $\omega(A^\#) = A$ for all $A \in \mathfrak{g}$ and $ad_g(R_g^*\omega) = \omega$.

*Let $VP$ denote the vertical bundle of $P$, i.e. the kernel of $d\pi: TP \to TM$.  An Ehresmann connection is a complementary subbundle of $TP$, i.e. a subbundle $HP$ of $TP$ such that $TP = VP \oplus HP$ as vector bundles over $P$,   which is invariant under the $G$-action: $R_g^* H_u P = H_{ug} P$.


There is a one-to-one correspondence between connection 1-forms and Ehresmann connections as follows.  Given a connection 1-form $\omega$, the kernel of $\omega$ is $G$-invariant and complementary to $VP$, so it defines an Ehresmann connection.  Conversely, given an Ehresmann connection $TP = HP \oplus VP$, let $q: TP \to VP$ denote the natural projection map (a morphism of vector bundles).  Note that the vector fields $A^\#$ define an isomorphism between $VP$ and the trivial bundle $P \times \mathfrak{g} \to P$, so composing $q$ with the projection $VP \cong P \times \mathfrak{g} \to \mathfrak{g}$ gives a map $TP \to \mathfrak{g}$, i.e. a $\mathfrak{g}$-valued 1-form.  This is a connection 1-form.
Unwinding these identifications, you could just as well use a third definition:


*A connection on a principal bundle is a Lie-algebra valued 1-form $\omega$ which satisfies $\omega(A^\#) = A$ for all $A \in \mathfrak{g}$ and whose kernel $H = \ker \omega$ is $G$-invariant.


This is your definition.  From here you can deduce that $H$ complements $VP$ in $TP$, (or that $\omega$ is compatible with the adjoint action of $G$) and hence connections in this sense are the same as the objects defined in the previous two definitions.
