I am studying on my own from do Carmo's Riemannian-geometry text and in chapter 3, the author introduces 'normal neighborhood of a point in Riemannian manifold $M$ and normal neighborhood of $0 \in T_{p}M$ and then introduces normal coordinates around a point $p \in M$.
And then in subsequent topics the author frequently uses normal neighborhood, normal ball etc.in proofs.
Now, what I don't understand is, why do we need these special type of coordinate chart around a point and special type of balls?
It would be great if someone could refer me to some source or maybe explain here the motivation behind introducing such notions which is not clear just by reading the text.
Thank you!
The point here is that it is very difficult to find coordinates adapted to a Riemannian metric. Since curvature is a local invariant, you simply cannot find coordinates in which a metric has a prescribed form. Normal coordinates is about the best that on can do in that direction. They are adapted to the metric in the sense that in the central point the metric is approximated by the flat metric to first order (which is the best that you can do). However, they are well adapted to geodesics through the central point and hence to distances to that point. This is about as good as it gets.