I believe that is the Einstein's convention of writing sum.
I started studying Riemannian geometry a day ago and in my book it says you form the following bias coordinates in the tangent space.
And if you have a vector
What I don't understand is how can you make an operator in biasing coordinates. Does the above imply that i.e.
How are you defining the tangent space?
If the tangent space at is as an (equivalence class of) curve(s) , , which act on a function as , then you can chuse a coordinate chart on and rewrite where acts on a function as . Therefore the set of these is spanning and simple examples show it is linearly independant...