Unique separation of the vertical and horizontal spaces?

  • #1
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?
 
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  • #2
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Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
 
  • #3
lavinia
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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?

The horizontal space at a point in the principal bundle is just the kernel of the connection 1-form. The tangent space splits into a direct sum ##K⊕V## where ##K## is the kernel and ##V## is the vertical space. Connection 1-forms differ by their kernels since their restrictions to the vertical spaces are all the same.

In general two linear maps can agree on a subspace but have different kernels. The vector space is always isomorphic to the direct sum of the image and the kernel of the linear map.
 
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  • #4
Thanks very much!!!
 

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