Let $M$ be a 1-dimensional manifold and let $E$ be its vector bundle. I want to show that every connection $D$ on this vector bundle is flat.
A connection $D$ is flat means that we have $$D_v D_w s-D_w D_v s -D_{[v, w]}s=0 $$ for all vector field $v, w$ and section $s$.
I am not matured in differential geometry and I don't know where to start. I don't have any idea how to use the condition that $M$ is -dimensional.
I appreciate any help.
Also, do you have any geometric intuition when you read this kind of sentence? If so, how did you learn that intuition?
You only have one dimension for your vector fields. Write $v=a\,\partial/\partial x$ and $w=b\,\partial/\partial x$, and compute. (My personal intuition comes from thinking of curvature of a connection as a $2$-form, and the only $2$-form on a $1$-dimensional manifold is $0$. Zero curvature $\iff$ flat.)