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? ![e=mc^2]

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?

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?

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? ![e=mc^2]