Question about connections on the dual bundle.

Question

Let $E \to M$ be a vector bundle with connection $\nabla$. Extend $\nabla$ to $E^*$ and $E^* \otimes E$ in the regular fashion. Is $\text{Id} \in E^* \otimes E$ necessarily parallel?

Answer 1

Let $e_1, \cdots, e_n$ be a local basis of $E$ so that $\nabla e_i (x) = 0$ at $x\in M$. Using $1.$, the dual basis $e_1^*, \cdots, e_n^*$ also has $\nabla e_i^* (x) = 0$. Using this basis, we have locally $$\text{Id} = \sum_{i=1}^n e_i^* \otimes e_i,$$ thus $\nabla \text{Id}(x) = 0$ by $2.$. Since $x\in M$ is arbitrary, we have $\nabla \text{Id} = 0$.