I'm confused about a specific issue that I have with the Einstein notation (for tensor fields on manifolds).
I want to write the following thing:
Let $X$ be a smooth manifold. Choosing local coordinates $(q^1, \dots, q^n)$ on $X$ defines natural coordinates $(q^1, \dots, q^n, p^1, \dots, p^n)$ on $T^{*}X$: the point with coordinates $(q^1, \dots, q^n, p^1, \dots, p^n)$ is the covector $p^i dq^i \in T^{*}X$.
As you see, there is a problem with $p^i dq^i$: both indices are up. A such covector should be written $p_i \, dq^i$. On the other hand, coordinate functions are written with upper indices, so as coordinate functions on $T^{*}X$ it seems like $(q^1, \dots, q^n, p^1, \dots, p^n)$is correct.
What is going on? What should I write?