Fibre Bundles Part 2

Problem 5.36 Given a linear connection $\nabla$ on the $C^{\infty}$ manifold $M$, one defines the conjugate or opposite connection $\widehat{\nabla}$ on $M$ by $$ \widehat{\nabla}_{X} Y=\nabla_{Y} X+[X, Y], \quad X, Y \in \mathfrak{X}(M) $$ (i) Prove that $\widehat{\nabla}$ is a linear connection. (ii) Compute the local components $\widehat{\Gamma}_{j h}^{i}$ of $\widehat{\nabla}$ in terms of the components of $\nabla$.

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 $$ L_{X} \circ \nabla_{Y}-\nabla_{Y} \circ L_{X}=\nabla_{[X, Y]}, \quad X, Y \in \mathfrak{X}(M) \text {. } $$

Problem 5.38 Let $M$ be a $C^{\infty} n$-manifold endowed with a torsionless linear connection. Prove that in a system of normal coordinates with origin $p$, all the Christoffel symbols at $p$ vanish.