Let \(M\) be a smooth manifold and \(F\in \Gamma(T^{(k,l)}TM)\) a smooth tensor field . Then applying the total covariant derivative on \(F\) \(n\)-times, we obtain a smooth \((k,l+n)\)-tensor field \(\nabla^nF=\nabla(\cdots \nabla F)\). We write \(\nabla^n_{X_1,\ldots,X_n}F(\ldots)=\nabla^2F(\ldots,X_n,\ldots,X_1)\). Note, that the convention is, to apply \(X_1,\ldots,X_n\) in the reverse order.
Remark
- \(\nabla^n_{X_1,\ldots,X_n}F\neq \nabla_{X_1}(\cdots \nabla_{X_n}F)\) (as in the second covariant derivative case)
- High order covariant derivatives are not symmetric in all arguments.