Let $(M,g)$ be a Riemannian manifold and $X\in\Gamma(TM)$ a vector field. We choose local coordinates $x^{i}$ on $M$. My question:
Does the notation $X_{;ij}^{k}$ mean the components of $\nabla_{\partial_{j}}\nabla_{\partial_{i}}X$ or of the double covariant derivatice $\nabla^{2}_{\partial_{j},\partial_{i}}X:=\nabla_{\partial_{j}}\nabla_{\partial_{i}}X-\nabla_{\nabla_{\partial_{j}}\partial_{i}}X$?
I would have assumed the latter, since for example also the notation $\nabla_{j}\nabla_{i}X^{k}$ which can be found in many physics books refers to the components of $\nabla^{2}_{\partial_{j},\partial_{i}}X$. Also in Lee's book on Riemannian manifolds, it seems to be used for the latter (see Example 4.22). However, on wikipedia (section "Notation"), it seems that they state that it used for the former.