Local Expression of Connections in Tensor Bundles

Let \(M\) be a smooth manifold with or without boundary and \(\nabla\) a connection in \(TM\). Given a local frame \((E_i)\), it dual coframe \((\varepsilon^j)\), and \(\{\Gamma_{ij}^k\}\) the connection coefficients of \(\nabla\). Let \(X\) be a smooth vector field and \(X^iE_i\) its local expression.

If \(F\in \Gamma(T^{(k,l)}TM)\) is a smooth tensor field of any rank, with the local expression

\begin{equation*} F=F^{i_1\ldots i_k}_{j_1\ldots j_l}E_{i_1}\otimes \cdots \otimes E_{i_k}\otimes \varepsilon_{j_1}\otimes \cdots \otimes \varepsilon_{j_l}, \end{equation*}

then the covariant derivative is given locally by

\begin{equation*} \nabla_XF=\Bigl(X\bigl(F^{i_1\ldots i_k}_{j_1\ldots j_l}\bigr)+\sum_{s=1}^{k} X^mF^{i_1\ldots p\ldots i_k}_{j_1\ldots j_l}\Gamma_{mp}^{i_s}-\sum_{s=1}^{l} X^mF^{i_1\ldots i_k}_{j_1\ldots p \ldots j_l}\Gamma_{mj_s}^{p}\Bigr) E_{i_1}\otimes \cdots \otimes E_{i_k}\otimes \varepsilon_{j_1}\otimes \cdots \otimes \varepsilon_{j_l}. \end{equation*}