Why Are Connection 1 Forms Essential in Principal G-Bundles?

[hideelo]

Suppose I had a principle G-bundle (P,π,M) and I wanted to define for each p in P the vertical and horizontal subspaces of $T_pP$. If I considered any point p, I can consider $\pi_* : T_p P \rightarrow T_pM$. Why can't I divide $T_pP$ into the kernel of $\pi_*$ which I will call my vertical and the rest will be my horizontal? Why do I need connection 1 forms?

 

[lavinia]

Suppose I had a principle G-bundle (P,π,M) and I wanted to define for each p in P the vertical and horizontal subspaces of $T_pP$. If I considered any point p, I can consider $\pi_* : T_p P \rightarrow T_pM$. Why can't I divide $T_pP$ into the kernel of $\pi_*$ which I will call my vertical and the rest will be my horizontal? Why do I need connection 1 forms?

The direct sum decomposition of a tangent space at a point in the principal bundle is not determined by the differential of the bundle projection map. Think about what you really mean when you say "the rest".

 

[hideelo]

The direct sum decomposition of a tangent space at a point in the principal bundle is not determined by the differential of the bundle projection map. Think about what you really mean when you say "the rest".

I guess I wasnt thing that in order to define orthogonal, I need some inner product structure, just being a vector space isn't enough.

Thanks

 

[lavinia]

I guess I wasnt thing that in order to define orthogonal, I need some inner product structure, just being a vector space isn't enough.

Thanks

You do not need an inner product, just a direct sum decomposition of the tangent space to the principal bundle into the tangent space to the fiber and a subspace that projects isomorphically onto the tangent space to the manifold. For a connection, one also requires the horizontal spaces to be invariant under the action of the structure group. This is why a G-invariant 1 form with values in the Lie algebra of the structure group is used. It gives you all of these properties in one fell swoop. The horizontal space at a point is its kernel and G-invariance of the horizontal spaces follows from the G-invariance of the 1-form. In fact, the two, the 1-form and the horizontal spaces - are equivalent. That is: the one determines the other.