I am reading P. Li's lectures on Geometric analysis. On page 14, the author defines the second covariant derivative as follows:
Let $f$ be a smooth function on $M$. $\omega_1, \cdots, \omega_n$ be a local orthonormal basis of $T^*M$ around a fixed point $p$. And $d\omega_i=\sum_j\omega_{ij}\wedge\omega_j$. Then $$df=\sum_i f_i\omega_i$$ THen the author give the definition: $$ f_{ij}\omega_j=df_i+f_j\omega_{ji}. $$
My question is how to understand this definition compare with the usual one, i.e. $$ \nabla^2_{X,Y}f=XY(f)-\nabla_XY (f) $$
Also the third order covaiant derivative defines in the similar manner: $$ f_{ijk}\omega_k=df_{ij}+f_{kj}\omega_{ki}+f_{ik}\omega_{kj} $$ The similar question for this expression, how to understand these indices?
Thank you for any detailed answer, I am kind of afraid of this local calculations compare with the ususal global definition. However, in most papers, the local calculations are more common. So I want to figure it out clearly.