Chart Transition Map for an Atlas With One Chart

Question

I'll probably get negative reputation points for this but here goes. I've been asked to consider an atlas with a single chart $\{(U_i,x_i)\}$ on a manifold $M$ of dimension 1 (I won't say exactly what the chart is) and I need to talk about the chart transition map. My question is, what is the correct formalism for this notion? It seems silly to consider $x_i\circ x_i^{-1}: \mathbb{R}\to \mathbb{R}$. I mean, in this case, you aren't really transitioning to another chart image of a subset of the manifold which seems contrary to the notion of a chart transition map. So, to be concise, my question is, can I consider $x_i\circ x_i^{-1}: \mathbb{R}\to \mathbb{R}$ to be a chart transition map if the map takes you to where you started?

Answer 1

If you have an atlas $A = \{(U, \phi)\}$ i.e with one chart then the transition map is the identity. The idea behind the transition map is to allow one not to lose information when using different charts on the same domain. You can see this in the fact that we require both $\phi_i \circ \phi_j^{-1}$ and $\phi_j \circ \phi_i^{-1}$ to be smooth i.e the transitions are diffeomorphisms.