Curvature of Connection [exercise proofcheck]

connectionscurvaturedifferential-geometryvector-bundles

I tried to solve the following question, but I get the feeling I am doing a few things wrong, but cannot really find out what.

So the exercise is:
Let $\alpha, \beta \in \Omega^{1}(M)$ for a manifold $M$. And let a connection (''locally'') be given by $\nabla = d + A$ where $A$ is the following matrix: $$ \begin{pmatrix} 0 & \alpha \\ 0 & \beta \end{pmatrix} .$$

Compute the curvature of the connection over the trivial rank 2 vector bundle over M, and state conditions for this connection to be flat.

I've got as far as simply using the definition of curvature (and flatness) given: $\kappa_\nabla = dA + A \wedge A = 0$ to find:
$$ \kappa_\nabla = \begin{pmatrix} 0 & d\alpha \\ 0 & d\beta \end{pmatrix} + \begin{pmatrix} 0 & \alpha \wedge \beta \\ 0 & 0 \end{pmatrix} .$$

Then I'd say the conditions are: $d\alpha + \alpha \wedge \beta =0$ and $d\beta = 0$. A section $s$ of the trivial vector bundle of rank 2 is nothing else than an element $s \in C^{\infty}(M)^{\oplus2}$ so $s:= (f_1,f_2)$. We see then that:

$$ \kappa_\nabla(s) = \begin{pmatrix}f_1 d\alpha \\ f_2 d\beta \end{pmatrix} + \begin{pmatrix} f_1 \alpha \wedge \beta \\ 0 \end{pmatrix} .$$

I feel like I've gone wrong because:

  1. I haven't used the fact that this is the trivial rank 2 vector bundle over M. Perhaps only in implicitly assuming the local expression of $d+A$ is valid everywhere.
  2. This feels like too little work, can the conditions for example be worked out more explicitly?
  3. The trivial vector bundle is flat, so shouldn't every connection be the flat connection?

Best Answer

Your work is good.

  1. the definition of the connection is global instead of local, and the chosen frame is implicitly assumed to be $s_j\colon M \to M\times \Bbb R^2$ given by $s_j(x) = (x, e_j)$, $j=1,2$, where $\{e_1,e_2\}$ is the standard basis of $\Bbb R^2$.
  2. the curvature is a tensor, so you don't need to say what does it do to an arbitrary section to conclude it must vanish -- it suffices to see what happens with the global frame from 1.
  3. absolutely not, flatness is always relative to a connection; you can't say that the "trivial bundle is flat" if you have not specified a connection.
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