Problem
Problem 5.37 Let $\varphi: M \rightarrow M^{\prime}$ be a diffeomorphism. Given a linear connection $\nabla$ on $M$, let $\nabla^{\prime}=\varphi \cdot \nabla$ be defined by
$$
\nabla_{X^{\prime}}^{\prime} Y^{\prime}=\varphi \cdot\left(\nabla_{\varphi^{-1} \cdot X^{\prime}} \varphi^{-1} \cdot Y^{\prime}\right), \quad X^{\prime}, Y^{\prime} \in \mathfrak{X}\left(M^{\prime}\right)
$$
Prove:
(i) $\nabla^{\prime}$ is a linear connection on $M^{\prime}$.
(ii) If $\varphi_{t}$ is the flow of a vector field $X \in \mathfrak{X}(M)$ such that $\varphi_{t} \cdot \nabla=\nabla, t \in \mathbb{R}$, then
$$
L_{X} \circ \nabla_{Y}-\nabla_{Y} \circ L_{X}=\nabla_{[X, Y]}, \quad X, Y \in \mathfrak{X}(M) \text {. }
$$