Let L=R2×U(1) be the trivial U(1)-bundle over R2. Define a connection ∇=d+A where A=fdx+gdy is an R valued 1-form on L. That is, ∇ gives a distribution H on L - the horizontal distribution.
The distribution H is obtained as the graph of −A as a linear map from R2→R. A horizontal lift ˜X of a vector field X on R2 is given by ˜X=(X,−A(X)).
Let α be the projection onto the vertical direction on L i.e. ker(α)=H, and define the curvature 2-form Ω∇ of the connection ∇ by
Ω∇(X,Y)=α([˜X,˜Y])
The following is expected to be true
Ω∇=−dA?
Here is my confusion:
Let z be the local vertical coordinate, X=∂x and Y=∂y. Then ˜X=−f∂z+∂x and ˜Y=−g∂z+∂y. And
−dA(X,Y)=−X(A(Y)+Y(A(X))+A([X,Y])=−∂xg+∂yf
while
[˜X,˜Y]=(f(∂zg)−g(∂zf)−∂xg+∂yf)∂z
therefore
Ω∇(X,Y)=α([˜X,˜Y])=f(∂zg)−g(∂zf)−∂xg+∂yf.
Where is the mistake? Thank you.
This post imported from StackExchange Physics at 2016-07-29 20:36 (UTC), posted by SE-user Student