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 \(\nabla^nF=\nabla(\cdots \nabla F)\).
Remark
- \(\nabla^n_{X_1,\ldots,X_n}F\neq \nabla_{X_1}(\cdots \nabla_{X_n}F)\)