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I have trouble understanding how a connection one-form can separate and tangent space##{T_u}P## 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?

To be more specific, for two connection one-forms ##{\omega _1}## and ##{\omega _2}## that satisfy condition 1 and 2, do they have the same kernel automatically?

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?

To be more specific, for two connection one-forms ##{\omega _1}## and ##{\omega _2}## that satisfy condition 1 and 2, do they have the same kernel automatically?

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