I am reading John Lee's Riemannian Manifolds, and am having trouble with exercise 4.3.
In order to solve the exercise, you must show that the following definition of a connection on a tensor bundle commutes with contraction.
$(\nabla_X(F))(w^1, ..., w^l, Y_1,...Y_k)=X(F(w^1, ..., w^l, Y_1,...Y_k))-\sum_i F(w^1, ...,\nabla_X w^i,... w^l, Y_1,...Y_k)-\sum_j F(w^1, ..., w^l, Y_1,...,\nabla_X Y_j,...Y_k)$
I was able to prove that the operators commute for tensor bundles in $T^1_1$ but am having trouble for other cases.
How do you prove this for general tensor bundles?
As the tensor $F$ can be locally written as a sum of tensors of the form $Y_1 \otimes \cdots \otimes Y_{\ell} \otimes \omega^1 \otimes \cdots \otimes \omega^k$, we will suppose that $$ F = Y_1 \otimes \cdots \otimes Y_{\ell} \otimes \omega^1 \otimes \cdots \otimes \omega^k $$ Let us say that we contract the indices $\ell$ and $k$, then $$ \operatorname{tr} F = \operatorname{tr} (Y_\ell \otimes \omega^k) Y_1 \otimes \cdots \otimes Y_{\ell-1} \otimes \omega^1 \otimes \cdots \otimes \omega^{k-1} = \omega^k(Y_\ell) Y_1 \otimes \cdots \otimes Y_{\ell-1} \otimes \omega^1 \otimes \cdots \otimes \omega^{k-1} $$ By the product rule (when solving Lee's exercise, you first need to establish the product rule (c) before showing that the connection commutes with contractions), and after some simplifications, we obtain $$ \operatorname{tr}\nabla_X F - \nabla_X\operatorname{tr} F = \left( \omega^k(\nabla_X Y_\ell) + (\nabla_X \omega^k)(Y_\ell) - X(\omega^k(Y_\ell))\right) Y_1 \otimes \cdots \otimes Y_{\ell-1} \otimes \omega^1 \otimes \cdots \otimes \omega^{k-1} $$ which is zero by the definition of $\nabla_X$ for covector fields (formula (i) in Lee's book).